What kind of language would a sailor use to actually curse, i.e. how would one literally curse like a sailor?

This is a great question, and there are a few levels of answer to it. Before I start I want to distinguish between types of "bad language" -- roughly, we can draw a distinction between profanity (taking Gods's name in vain, or cursing God -- often called "blasphemy"); obscenity (sexual or scatological language); and vulgarity (coarse or crude language that might not rise to the level of vulgarity).

In the time period you picked (early 1700s), England was still coming out from the long shadow of the civil wars and counter-revolutions of the 17th century, when the fleet had often been under the control of or in opposition to a religiously-minded military dictatorship. The Puritans had attempted to impose strict religious order on the navy, with mixed success; captains went to sea with orders to not only keep religious observations, but also to suppress blasphemy. Their record in doing so was mixed at best, but William Penn, one of the three "generals at sea" in the 1650s, suggested flogging or a 5 shilling fine for each episode of blasphemy. Rules against profanity and vulgarity seem to have been regarded as unenforceable.

However, there was a different insult entirely, among sailors, that would anger even the most hard-bitten mariner. I'll quote myself at some length from a previous answer:

Turning to an illustration of how much men valued their status as seamen would take us to the mutiny of the HMS Hermione, which is the bloodiest mutiny of the British fleet in its history.

Mutinies, at least before the Napoleonic period, were actually more in the nature of popular demonstrations or workers' strikes, where men would send a letter of grievances to the captain or a higher authority, and were often provoked by suddenly changing officers or captains or a lack of what men considered their perquisites -- tobacco, beer, victuals, etc. The mutiny on the Hermione was completely different.

HMS Hermione was a frigate with a short but decently distinguished naval record, which had been in the West Indies from 1793, at the start of the French Revolutionary wars, and participated in several small engagements. When her captain died of yellow fever, he was replaced by a man named Hugh Pigot, who had used patronage to be quickly promoted post-captain (he was 28 at the time of the Hermione mutiny). Pigot was known as a liberal flogger -- while flogging was a normal punishment in the Royal Navy, he managed to flog 85 men of his previous crew -- about half -- and two so badly they later died from their injuries.

Pigot continued this type of discipline on board Hermione, and made two errors in particular that led to the mutiny. In the first, he found fault with a knot tied by a sailor and blamed that sailor's midshipman for the problem (midshipmen at this point commanded divisions of sailors, with supervision). He asked the midshipman, David Casey, to apologize to him on his knees on the quarterdeck; when Casey refused this as being a type of debasement unfitting for a gentleman, Pigot disrated him and had him flogged. This deeply upset the sailors Casey had been in charge of, and they began to talk of mutiny -- disrating a midshipmen could be done under some circumstances, but the obvious intent to humiliate upset the social order (such as it was) that normally existed on the ship, or at least would have existed on a well-run ship.

Pigot also developed a taste for flogging the last men down from the masts, which was seen as not only arbitrary but unfair, as the last men down were usually the men who went out to the very ends of the yardarms when making sail or reefing sails. On Sept. 20, 1797, a squall struck the ship, forcing it to reef sail, and Pigot gave his customary flogging order. Three topmen, rushing to get down, fell and were killed (one struck and injured the master). Pigot's reaction was to order "throw the lubbers overboard" -- "lubber," as in "landlubber," being the worst insult in a sailor's vocabulary. When two topmen complained, he had them flogged, and flogged the rest of the topmen the next day.

This seems to have ben the final straw for the Hermiones.

On the evening of Sept. 21, several sailors who were drunk on stolen rum overpowered the Marine sentry outside of Pigot's quarters, forced themselves inside, and hacked at him with knives and swords before tossing him overboard, possibly still alive. The sailors -- about 18 total -- then hunted through the ship and killed eight other officers, a clerk, and two midshipmen, sparing some warrant officers (including the sailing master, who could navigate the ship). The mutineers turned Hermione over to the Spanish, who took her into service as a frigate, manned by 25 of its former sailors under heavy guard.

The British reaction to the mutiny was to hunt down and try former sailors; they eventually captured 33, of whom 24 were hanged and gibbeted. Hermione sat in the harbor of Puerto Caballo for two years, until boats from HMS Surprise cut her out with heavy casualties on the Spanish side. The ship was renamed Retaliation and later Retribution.

The cause of the mutiny, and the violence that ensued, is almost certainly the result of major and repeated breaches of the implicit social contract on board ship by Pigot. His repeated insults to seamen and arbitrary punishments certainly set the stage for the mutiny, but his insult to their professional competence seems to be what caused it to break out in such violence.