Haven't been to any in Alabama or Georgia, but we've been to plenty of National Forest Service campsites in NC, Virginia, Pennsylvania and New Hampshire. They're usually rustic, some have dump stations, more rarely electric and water at the site. Not quite boondocking but close.
It's not easy to find RV accessible dispersed campsites in eastern National Forests, there are some in the Asheville NC area near North Mills River, also on Forest Service roads near the Cradle of Forestry near Brevard (found these using a Delorme backroads map, nowadays a lot easier with Google Earth!) Also some forests and gamelands have hunt camps that can be used all year.
Try looking at this site
, this site
or this site
for some ideas. We had to dig for a while to find the ones we found, but have thoroughly enjoyed the ones we've been to, plenty of hiking/fishing/wilderness within walking distance.