I looooove Hocking Hills! :)
Try the campground called Three Reasons. We camped there last Easter - well-kept, nice stalls, electric hookup, water, and good trail access. Close to Airplane Rock, so it's not a far ride to get to the rappelling area/Tabletop Rock, Conkles Hollow, Rock House, Old Man's Cave, Artesian Well, and other popular sites.
They also have a cabin (maybe two) that can be rented right at the camping area, and I think a bathroom/shower house as well. Our trailer has a toilet and shower and I never went into the campground's facilities to see how it looked, but I never heard any of the other campers complain about it.
According to some reviews, the owner of the place is notorious for being difficult to get in touch with for checking in and out. That wasn't my experience - she didn't come around visiting all the sites like some campground owners, but she wasn't unfriendly.
I've heard about the place near the state campground - some friends camped there last year. Some said it was "okay," some were not so thrilled. Apparently there were problems with the electric hookups and they kept losing power.
There used to be a place called Palmerosa but I think it closed now - that's where I stayed the first time I camped there with horses. They had a lot of "year-round" site renters who took most of the better sites, built porches and corrals and things for themselves. The other sites were just kind of "wherever you can park near an outlet."
Enjoy your time there - I've been to places with MORE riding (as far as trails/miles) but IMO it's hard to find a prettier
place than Hocking Hills. Reminds me a lot of Shawnee, IL but on a smaller scale (and a shorter drive).