You might try Lake Juliette, Lake Jackson or even Lake Tobesofkee. They are Georgia Power lakes that have a few camping spots that aren't too expensive. High Falls has lake and river camping, but check their website, I think there's a 10HP engine limit. It's more for kayaks and canoes.
www.recreation.gov will give you a lot of sites, but you many need to travel a couple of hours. West Point lake is on the GA/AL border, but it's a larger lake with plenty of fish.