Cornwallis had one of the armies in the South with Alexander Stewart commanding another in South Carolina around Charleston. The lack of a foothold in the states and resilience of the American army helped push the British surrender and evacuation. British forces held New York City, Charleston, Georgia, and other places but could not fully control full regions. Even though the British army held Charleston, they did not have control of the rest of South Carolina. Slowly, the British forces in SC trickled back to the Charleston area after evacuating other posts such as Camden and Ninety-Six, and losing forts such as Fort Motte. The Battle of Eutaw Springs put an end with British forces trying to gain more control of South Carolina as the army would remain there until total evacuation. Even the great victories of Camden and Charleston did not seem to quell the continuing operations of the American armies. In addition, while Henry Clinton commanded the entire British army, he had distanced himself from Cornwallis after May 1780. Clinton had a disagreement with Cornwallis and for a time he would correspond with Cornwallis for updates they became sporadic by mid-1781. Clinton remained the commander in chief until 1782 when Guy Carleton took over, but the war had just about ended by then.
The British army after Saratoga had diminished substantially. Burgoyne's army became part of the Convention army. Not really prisoners of war but could not fight anymore in the war. That surrender took away 6,000 soldiers from the British army - a huge blow. Cornwallis' army numbered to almost 8,000. Clinton had a small force in New York and Stewart's army in Charleston had about 2,500. None of these armies had the capability to go against the larger American force. The British army had spread itself thin with the various theatres that popped up in 1778 including Anglo-French, Anglo-Spanish wars that made them fight not only in the Caribbean but the west including Louisiana and Kentucky (at least, what would become them). The surrender of Cornwallis' army spelled the end of the war effort in America. At that point, the American army had become a well disciplined force that showed no sign of shrinking. With the addition of Spanish and French forces, the British army had a lot to contend with and could not with the decreased amount of soldiers.
Often times, army leaders plan on crushing enemy forces and occupy the capital cities to win the war. The colonies remained decentralized and if one city fell, the rest would remain in resistance. The British army also failed to fully crush enemy combatants after a battle. Most militia units in the South had horses for retreating purposes and could swiftly escape the field. American forces also tended to regroup after battles and gain reinforcements, as well as relying on more guerilla tactics.
The 1781 campaign started with the battle of Cowpens where the British forces lost and began the Race to the Dan. This would in turn force Cornwallis' hand with the need to chase Nathaniel Greene or beat him to the Dan river. The lose of the battle also spelled disaster for the rest of the campaign as the British army had to leave supply wagons behind in order to march at a quick pace through North Carolina. It also halted his plans for controlling the rest of South Carolina. The long chase ended at Guilford Courthouse with Cornwallis battling a renewed American army. After suffering a Pyrrhic victory, Cornwallis decided to head to the coast in order to get reinforcements and supplies at Yorktown. Cornwallis also had some of the most experienced and seasoned regiments with him at the surrender including the 33rd, 71st, 23rd, the Guard infantry, the British Legion, and German battalions.
With morale crushed, Lord North proclaimed the war to be over and the administration switched from war to peace negotiations (and prisoner exchange). Ironically, the naval campaigns in the Caribbean had been quite successful but not enough to continue the war.
Matthew Spring, With Zeal and Bayonets Only.
John Buchanan, The Road to Guilford Courthouse.
Charles Ross (ed.), Correspondence of Charles, First Marquis of Cornwallis.
/u/GeneralLeeBlount has covered the tactical military reasons that continued fighting would have been problematic for the British, but you are right, it was conceivable that the British could have kept going. In fact, there actually were some smaller engagements after Yorktown, including the Battle of the Saintes, and the Siege of Fort Henry. But even in the best case scenario if the British had kept on fighting, it was not going to be over soon, and the British had their armed forces engaged in more than just the Revolutionary War at the time.
From a governmental standpoint, though, the British forces stopped fighting because there was no longer any political will to fight it. The warhawks in Parliament lost support, the majority was being threatened with losing control of the government, so instead of sending orders to engage in further conflicts with the Americans, the government began looking for a way to end the conflict.
News of the surrender reached London on November 25, 1781. When Prime Minister Lord North received the news of the defeat at Yorktown, he is reported to have responded with, "O God, it is all over!"
The evening the news was delivered, Lord George Germain, the Secretary of State for the Colonies, who was in charge of the government's war effort, hosted a dinner party attended by Lord North and King George III to discuss what to do next. Germain and King George wanted to continue the fight, and Lord North "initially gave no public indication that the government had any intention of withdrawing from America." But he privately confided to friends and allies that he no longer saw a way to win it, given the tenuous hold he had in Parliament.
The King's Speech was delivered to the House of Commons two days later, which admitted to the defeat at Yorktown, but apart from that, was designed to rouse the public to continue the fight. But the opposition smelled blood. Opposition leader Charles James Fox countered that Lord North's government had conducted a disastrous war, the British public were against them, and it was time to end it if they wanted to avoid the public's wrath:
"...[T]he representatives of the people must recall to the ears of his Majesty's Ministers the disgraceful and ruinous measures that had brought us to this state. They must hear of them not only here, but...by the aroused indignation and vengeance of an injured and undone people, the ministers would hear of their ruinous measures at the tribunal of justice, and expiate them at the public scaffold."
A Tory ally of Lord North's, Sir Nathaniel William Wraxall, would later recollect that Lord North conceded this was true. The war had started with popular support among the British public, but repeated battlefront defeats had changed public opinion:
"True it is that ill success rendering [the war] at length unpopular, the people began to cry out for peace."
Historian Andrew Jackson O'Shaughnessy gives a pretty concise rundown of what happened next in The Men Who Lost America: British Leadership, the American Revolution, and the Fate of the Empire. "It was a testament to his popularity and his political skills," writes O'Shaughnessy, "that [Lord North] managed to sustain his government for over three more months."
Lord North attempted to negotiate with American diplomats Benjamin Franklin, John Jay, and John Adams for a peace that would keep the colonies under British control, but the Americans would accept no deal that did not recognize them as an independent country. Nor would Lord North commit to ending further military action until a deal was agreed to. According to O'Shaughnessy, North "was ultimately driven from power by his refusal to make a clear commitment to military withdrawal from America." North already had a slim majority, by barely a dozen votes, and by the early months of 1782, independent members of Parliament began to defect.
At the end of February 1782, General Henry Seymour Conway proposed a resolution that the war "no longer be pursued for the impracticable purpose of reducing the inhabitants of that country to obedience by force" and the resolution only failed by a single vote.
Two weeks later, on March 15, an independent MP proposed to censure the government for having "reduced the country from a state of glory and prosperity to calamity and disgrace." It only failed by nine votes.
On March 25, Charles James Fox called for peace with America and motioned for a vote of no confidence against Lord North because "our affairs were so circumstanced, that ministers must lose their places, or the country must be undone." The vote nearly passed.
With the prospect of another vote of no confidence that was sure to pass, two days later, on March 27, Lord North resigned. The Whig leader, the Marquess of Rockingham, then formed a government, and sent Richard Oswald to Paris in April 1782 to begin negotiating with the American diplomats.
Rockingham actually died a few months later, which sent Parliament into further turmoil, ultimately leading to North returning to power in an uneasy, and short-lived, coalition government in 1783 with his rival and chief war opponent Charles James Fox. But by then, the Treaty of Paris had already been drafted, and the coalition was formed with Fox as Secretary of Foreign Affairs, with North the head of domestic affairs, so peace negotiations continued unabated.
SOURCES:
The Men Who Lost America: British Leadership, the American Revolution, and the Fate of the Empire by Andrew Jackson O'Shaughnessy
A Diplomatic History of the American Revolution by Jonathan R. Dull
The Speeches of the Right Honourable Charles James Fox in the House of Commons, ed. by "A Barrister"
The Historical and the Posthumous Memoirs of Sir Nathaniel William Wraxall, 1772-1784 ed. by Henry Benjamin Wheatley
I only had the opportunity to briefly skim through some of these responses; however, a major point that I think needs clarifying is that while the British could have held major American ports, the largest among them being New York, for years, if not decades, after Yorktown, popularity for the war at home in Britain dropped dramatically in the late 1770s. Furthermore, in the eyes of the Empire, maintaining control of the Caribbean Islands, especially the sugar producing islands, was of the utmost concern due to both the tremendous profits generated and the power of "absentee planters" in British Parliament.
As Richard Buel Jr. argues in his book In Irons: Britain's Naval Superiority and the American Revolutionary Economy, the sheer size of His Majesty's Navy meant the British would be able to at least hold New York, if not capture a number of other American ports in the Mid-Atlantic and Southern colonies, even after Yorktown and probably hold these for decades. While possible, this avenue made little economic sense as the navy had a more important and lucrative role in protecting the plantation empire of the Caribbean. When you couple this with the idea the war in the colonies had been a disaster for the North administration (see both Mary Beth Norton's, The British-Americans: The Loyalist Exiles in England, 1774-1789 and Andrew Jackson O'Shaughnessy's, The Men Who Lost America: British Leadership, the American Revolution, and the Fate of the Empire) continuing to fight in mainland North America made little sense. Finally, it's important to note that while the British sugar producing islands had remained loyal, their loyalty was tenuous at best. If the British transferred forces from the Caribbean, who were not only fending of the French and Spanish but also suppressing potential slave insurrections where enslaved Africans greatly outnumbered white British colonists, they may have lost the the allegiance of island planters, many of whom held positions in Parliament. For more on this, see Andrew Jackson O'Shaughnessy's, An Empire Divided: The American Revolution and the British Caribbean.
TL;DR: While the British could have kept fighting in North America and perhaps held important ports on the eastern seaboard, it made little economic sense to do so and they risked losing even more Atlantic possessions if they did so.