Jefferson County Forest Phase IV, also known as the Horine section, offers both the type of camping you speak of (with special permit, I believe) and camping facilities with very little improvement (Water spigot and port-a-cans or pit latrines installed when the place was still a Boy Scout Camp). I haven't camped at Horine for around 10 years, but it was $5 a head per night back then.
There are some nice hiking trails of various difficulty, a couple of more or less logging roads, one graveled road to the old Horine family cemetery still active and on the property (rumored hauntings and a couple of rather odd phenomena surrounding that), a small pond that may or may not have fish but did have an impressive snake population when I was last there, and a couple of very nice pine orchards planted by old Doc Horine back in his day that are very pleasant to camp in so long as the winds are not high and there is no lightning. There is a trappers cabin on the property that might be available for use, though it was in pretty bad shape when I was there 10 years ago.
I spent many an enjoyable weekend there growing up. It is where I learned my tracking, stealth, and evasion skills. We had the benefit of several ex-military including one very grouchy special forces sgt. with a heart of gold as part of our adult leader corps. They seemed to always enjoy teaching us what they knew on subjects not quite endorsed by the BSA as appropriate for us to learn.
Something I learned early on is just how amazingly blind a person can be even when they are looking for something. I actually had people look right at me (step on me one time) and never see me. Funny thing, the only way I could have been anymore obvious in all of the cases except the stepping on one would have been to be screaming like a little girl and waving my arms like a flight deck controller. The most blatant failure was when I was leaning back against a tree (standing up) and smoking a cigarette. The search party passed within 20 yards of me, looked at me, and never even broke stride. They all swore (literal swearing) later that they never saw me.
It might not be what you are looking for, but it is a really nice place to camp. Just avoid major holidays. There are some large and rather rowdy groups that like to use the facility on holiday weekends.
