weather is anyones guess - and forecasts on the BRP or Skyline Drive are almost non existent.....just because Waynesboro/Luray/Front Royal don't have a forecast of snow/ice doesnt mean that the Skyline "Drive" isn't getting pounded.
I highly recommend
Andy Guest (aka Shenandoah River) State Park - a couple of miles south of Front Royal & the northern entrance to the Skyline Drive.
the accommodations at the park are great - 50A (you could need it if it gets cold) & Water (dump tank close to the camp slots) - nice large/private showers and washer/dryer facilities are close to the camp/slots too.
There are some (fairly new & fully equipped) FANTASTIC Cabins in the park and there are NEW
ZIP LINES in the park (canoeing, hiking, biking & fishing are also options).