Bulldog Park in Athens will rent sites outside of football season. I had checked on staying there a couple of years ago. I don't remember the price as we ended up in Savannah instead. I think there are some restrictions on age, size or something. My 28ft TT that is less than 5 years old made the grade, so I didn't worry about it much.
I am usually a state park person, but would suggest you consider staying in Athens, if you are going to be with your son most of the time. It will be nice not to drive back to RV late after dinner and other activities in Athens.
Edited to add "Go Dawgs! woof woof woof"