Yes, they did still claim the French throne.

While all the Kings of England after Edward III chose to claim it, the early Stuart Kings made it clear that they were not willing to engage in any battle for it (there are many reasons why, but it's worth noting that the early Stuart Kings were Scottish, and Scotland and France were old allies). During the interregnum, while Charles II was living in the French court as an exile, he diplomatically chose not to make reference to it, but he never agreed to cease using it if he was restored.

The old Pretender and Bonnie Prince Charlie both used the French title in conjunction with the British ones, even while receiving a pension from the French.

During the French Revolution, when the French Monarchy was abolished, France did make a request for it to be stopped, and one of George III's diplomats (James Harris, 1st Earl of Malmesbury) was sent to negotiate with them. He agreed that the King would stop signing documents as King of France, but those talks were interrupted and no further agreement was established.

It was not until the Act of Union joining Great Britain and Ireland that George III finally dropped the claim.