I would suggest St. Kitts except I don't think they have white sand
beaches. It was partly settled by the French, but I haven't eaten
ashore.

St. John has beautiful beaches and nice restaurants, but is pretty
expensive unless you camp at the National Park, and that may not be
your thing.

You might consider some of the places in Belize (like Caye Caulker, or
Ambergris) except that the beaches are very narrow.