Lake Powhatan is a great place. It is right off the Blue Ridge, just follow the signs to the North Carolina Arboritum. It is beautiful, wooded with great biking and hiking tails. Only 4 miles to the Biltmore Square Mall area. The two times I was there we were tent camping, pre Airstream days. I think they have some sites with hookups, ours had water close by but no electric. We prefer the "Big John" section. It is a NFS campground, rates are cheap and there are host on each loop and gate security.
Facility Details - Recreation.gov, Lake Powhatan (NC) - Recreation.gov
PS. Take a 1/2 day and see the Biltmore. Well worth it. You can save a few bucks if you get your tickets online ahead of time.