South Walton County in NW Florida might give you what you're looking for, never been but looks nice. I know they let residents drive on the beach so not sure if campers can, probably not. The state park below has all sorts of campsites and 3.2 miles of beach access.
Named for its tall dunes rising above the emerald waters of the gulf like a ship's sail, Topsail Hill Preserve State Park offers 3 miles of pristine white quartz beaches just minutes from Destin.
www.floridastateparks.org
2/3/20 - Updated link to the correct link to the State park, not sure what happened with the old link. I'll actually be going to this park and spending a couple of nights there the second weekend in April!