These are generalities, not specific to your area, but things to research, and of course several have already commented on these.
Any chance you are retired military? You will want to check Federal lands, including Forest Service even in the east, and definitely military lands if you are retired military. They have some options, I have heard. National Parks are more likely than other Fed lands to have only the specific camp sites. Once out west as people are saying, there are more choices for finding Federal open space without set campgrounds.
Just in case you do want to break down and try state parks, many state park are now on reservation systems where they tell you the sites and dates, and you can reserve specific sites online in advance. I know you said that is not your goal, but it beats pulling in at 8 pm wondering if anything is available.
If you really like that approach you mentioned, do some research and find hunting clubs, fishing clubs, other outdoors clubs, or even consider putting an ad out yourself to find a willing land owner. You know the problem in today's world though - fear of litigation. Don't forget counties. While I know of county campgrounds, I do not know of county wild lands, but it gives you something to research.
Good Luck, and report back!
