Oh howdy,
Before we look at Vichy French Resistance, we need to look at how they were structured prior to the landings.
Background
The French army in North Africa, nominally set by the Franco-German armistice of 1940 at 100,000, had been starved of replacements by the Axis so that it numbered only 74,763 under-equipped and relatively demoralized officers and men by November 1942. While French leaders were worried about an Allied invasion, they feared even more an invasion from the Spanish through Morocco. Decent support for the Allies too meant that some elements of the Vichy army were unreliable, but Vichy leaders could not come out and support the Allies just yet. There had yet to be a major invasion from the western allies and the Battle of Stalingrad was still in full tilt. Premature support would mean certain occupation from the Germans.
So what we have here is a force of dubious loyalty and of low morale without the proper equipment to maintain sustained resistance against an opposing force.
In addition, General Henri Giraud of France had escaped German captivity and been in secret negotiations with Eisenhower about flipping Vichy French loyalty to the Allies, so that was another factor Vichy French forces came up against prior to the invasion.
Let’s look at your questions.
How heavy was Vichy French resistance to the landings?
Here, we’ll take a look at several encounters between Allied soldiers and the Vichy French.
When the landings started, the Vichy commanders, Admiral Darlan and General Alphonse Juin were placed under house arrest by the Free French. Darlan and Juin attempted to radio to the Germans to ask for air support, but Allied code-breakers intercepted it. British Hurricanes destroyed 70 French planes on the ground. Some smaller suburbs in Oran contained some of the better equipped Vichy forces, but the Allies simply bypassed these local strongholds.
Off the coast of Morocco, much of the resistance came from naval action. French warships sallied out of Casablanca harbor to attack the U.S. armada. Four French submarines and six French destroyers were sunk or driven ashore by naval gunfire and by planes of the carrier Ranger. The heavy cruiser Primauguet went on something of a death run towards American ships but was badly damaged and run aground. The battleship Jean Bart managed to release 7 shells before it too ran aground.
The French defenders on shore were demoralized by the bombardment from U.S. ships standing off shore.
On November 8, late in the day, Darlan authorized Juin to seek a ceasefire with Allied forces. It took most of the day for the news to reach American forces in Morocco. The tally for the fighting was 1,346 casualties for the French, 1,225 for the Americans and 662 for the British. Afterwards nearly all resistance to the Allies in North Africa came from the Germans.
Did any Vichy Forces continue to fight with the Axis after Darlan gave his order to cooperate with the Allies?
We have to look at what Darlan did and the subsequent behavior of the French under his command.
The “Darlan Deal” was an agreement between Darlan and Eisenhower. This was controversial because Darlan had been a notorious collaborator with the Germans. Eisenhower wanted cooperation of the French, so he named Darlan as the commander-in-chief of all French forces in North and West Africa. The somewhat amazing part about this was that the forces under his command obeyed. There were only small pockets (dozens to maybe a couple hundred) of Vichy French (who were ideologues) that stayed with German forces. That effectively ended Vichy resistance to the Allied landings in North Africa.
It was really quite more anti-climactic than you would believe.
Sources
Atkinson, Rick. An Army at Dawn: The War in North Africa, 1942-1943.
Porch, Douglas. The Path to Victory: The Mediterranean Theater in World War II.