First of all, GO NOLES. Fellow Tallahasseean here.
There's A LOT of state parks in N-Fl, S-Ga. Three Rivers and Torreya are about an hour away each toward Chatahoochee. There's St. Andrews in Panama City. Jellystone Campground in Madison (there's a big slide into a lake at this campground). There's also a great state-maintaned campground near Lumpkin, GA called Florence Marina State Park. Just google it and you'll see. It has mini-golf, tennis courts, play grounds, etc etc etc and it's near Providence Canyon, affectionately known as the "Little Grand Canyon." PM me if you have more questions.