It would help to know your approximate route, and the area where you plan to stop each night.

If you can get as far as southern Ohio one night (not a horribly long drive from western VA), you might check out Old Man's Cave state park, near Logan, OH. There's an interesting gorge that can be a mile or so stroll, adjacent to the campground, to a 5 mile round trip to a waterfall.

Near Yellow Springs, Ohio (Springfield/Dayton area), about 5 miles south of I-70, is John Bryan State Park, with a lovely, little-used campground and another pretty gorge walk along a creek. Yellow Springs is a neat little town, with a couple of colorful local eateries due to Antioch College being there.

Near Crawfordsville, IN, about 20 miles south of I-74, you might check out Shades State Park. Again, there are about 15 miles of hiking trails, laid out in intersecting loops of about 2-3 miles each, so you could tailor an pleasant late afternoon stroll.

Coming across Kentucky, the Red River Gorge area in Daniel Boone National Forest (I think) and Natural Bridge State Park immediately adjacent are very scenic.

Mammoth Cave National Park might also be a pleasant stop.

If you're a Boy Scout, Scout leader (or maybe former Scout), local scout councils often have a camping area they use. You might contact them to see if they could be used.