Jekyll Island, near Brunswick, GA, has seasonal sites. There's not a lot to do there unless you golf or fish. Jekyll Island is an odd situation. The property was given to the state, but it is not a state or county park. It's owned by a separate entity. Some road signs refer to it as a state park, but you won't find it on the state park web site or publications. The island was formerly a winter home of the super rich.
Campground - Jekyll Island - Georgia's Vacation, Conservation and Educational Location
Manatee Hammock County Park, Titusville, FL, has seasonal sites.
Manatee Hammock Park Campground Home
We've stayed at both. They are a little cramped but okay. You won't be totally out of the cold because it does sometimes go below freezing in both places, but you won't have to worry about snow.