We have stayed at Grindstone (USFS) and liked it, Grayson Highlands SP in the area is good and great hiking trails right there, but no cell service last time I was there (have to drive down to 58). Hungry Mother State Park is about 40 minutes from Damascus off 81 and is nice.
There is a (newer) campground just outside Damascus. Haven't been to it, but have heard from somebody that has. Nice stay, but really just a parking lot with full hookups.
Laurel Creek RV Park Damascus, Va
Creekside RV park is right in town. Not a lot of room, tight spaces. But is right in town. If you eat/drink/take a shuttle to the trailhead, you don't need to touch the car. I think they are self-contained rigs only.