Really cool campground located right on the New River, located at Foster Falls. The river sounds amazing at night. You have to walk everything into the campground from the parking area, but it's not a long walk so you can still bring all of your "car camping" luxuries. There are no RV sites and there are no bathhouses, only pit toilets which can be a little funky, but it is what it is. Just a short walk from the campground, you can explore the area which includes stables, a wildlife center, visitor center with local artisan goods (we got some local teas, stevia, and handmade jewelry/ornaments as gifts), an abandoned hotel/grounds (you can check them out from the outside), an old train car, and much more. Pet-friendly. 

Sites 2, 6, and 9 would be my personal picks for next time. If you have a large group, G17 is HUGE. You can also use this campground for day-use, picnicking, etc.

Shot Tower State Park isn't far from here either, so definitely check that out while you're in the area. Millrace Campground is located within the New River State Park. We went in October and the colors were really starting to change.