I know you want Ga, NC, and Tn, but let me also throw in Natural Tunnel SP in Virginia. It's only 17 miles north of the Tn line and we are around 4 hrs from Columbia, SC. It is straight up I-26N, which then turns into Rt 23N. It's 1 mile off of Rt 23, so all good, nice beautiful 4-lane highway the whole way.
Natural TunnelThe campgrounds (2 of them) offer water/electric with a dump station within the park. Lover's Leap Campground is set up for larger RVs and has both pull-thru site (site #13) with a buddy site (site #14), but mostly back-in sites. They range from 45 ft long to 85 feet long and more provide some semblance of privacy as they face the wooded areas. There are lots of trails through mountainous areas, there is the actual Natural Tunnel that is so large that trains traverse through the tunnel daily. There is a large swimming pool with water slides and diving boards; they offer canoeing kayaking and canorkeling (canoe trip with snorkeling along the way) trips as well as caving excursions to nearby caves. All of these are guided trips. They also offer a guided trip to the Devil's Bathtub (see facebook or google it for some beautiful pictures).
We camp-host there two months each summer, so feel free to PM me if you have specific questions about the park.
I can guarantee that you and especially the kids will have a great time.