We just got back from our trip and ended up camping at the Low Water Bridge Campground, about 10 miles south of Front Royal, off Rt. 340, right on the Shenandoah River. Even though we opted for May rather than April, it was still too chilly (and foggy) in the National Park.
Although electric hook-ups were available, this is a "campground" rather than RV park. We liked it. Also, did a shuttled canoe trip on the river, available from one of two outfitters right next door.
Thanks to all for suggestions.