Don't know about that Georgia stretch, but Jonathon Dickinson State park is pretty close to West Plam Beach.
Jonathan Dickinson
It has two campgrounds; the one down by the river has more privacy, shade and is quieter. The one up by the highway is new with very little shade, but has sewage connections for long stays.
The local towns (Tequesta and Jupiter) are nice, and there's some great beaches nearby.
EDIT: now that I think about it, there are also Florida State Parks up near the Georgia border and Jacksonville. We think they're wonderful and spend a good part of the winter drifting from one to another. But don't forget the possibility of hurricanes in September.