Southeast resident here. I know you are looking for free options, but if you are willing to do some research you can find some pretty great extremely cheap option. Your local library has basically a phone book of all the campsites in america. We took it on a 7-week roadtrip. If the idea is less of a road trip, more of living as cheaply as possible wherever you can familiarize yourself with that series of books 'best campsites in . . ." "Best hiking trails in . . ." They have some pretty great recommendations.
For example, Red River Gorge in Kentucky offers super reasonable passes like 60 bucks if memory serves for the entire summer that lets you park and camp anywhere in the park for as long as you like. There's no picnic benches or RV hookups but it is cheap. Is that what you are looking for?