If you don't have an RV park in mind for Savannah and you can drive another hour, we like Coastal Georgia RV Park in Brunswick GA. Paved pull thrus. In Florida if you look during the week you might be able to find openings for a few days in state parks. You may have to move around more. On weekends or holidays, try campgrounds near the freeway that caters to overnighters. (like KOA's).
If you have time, I would stop in St Augustine. Lots to see and do there. Hop on and hop off trolley all day pass. Try Anastasia State Park. Definitely see Cape Canaveral. Jetty Park is a good place to stay but might already be booked. Of course Key West. Enjoy your trip!