Candy Hill (off I-81, near Winchester) is open year round. Big pull-through sites. Stayed there once. It was a decent place to stay, but didn't really WOW me.
Candy Hill Campground
It's a little further south, but you could try North Bend Park (border of VA and NC on Kerr Reservoir) which is only about 17 total miles out of the way when going down I-95 south (take I-85 south at Richmond and then cut back to I-95 using Rt. 1 south of the park). It's open year round, with really cheap sites and first-come and limited service during the off-season. We stayed there once on a trip to FL ourselves -- just try getting out of the DC area during rush hour on a Friday and see if you get any further than the border before you need to stop for the night. Recreation.gov says it's closed, but it's actually not, just the reservation service is closed. Call to verify, however.
Facility Details Booking Window - NORTH BEND PARK, VA - Recreation.gov