Not in VA, but I think Patapsco in MD opens in April. The Hilton tent area is great with kids - close to bathhouse (running water, lights, hot showers), small playground nearby, no noisy campers, and a big playground made of giant tires close by in the rest of Patapsco.

May be a bit further than what you specified but Douthat State Park west of Charlottesville was very nice.

Bull Run Campground
Pohick Bay Park
Prince William Forest
Westmoreland Park

Shenandoah state park?

Looking for a somewhat local (within ~3hr drive) place to go camping in mid-late April. Most Shenandoah campgrounds don't open until May and the one that is open (Big Meadows) is already full. We'll have a toddler so somewhere with bathrooms/running water is preferred (e.g., we like George Washington National Forest but that may be too "roughing it" right now). We did Cunningham Falls in MD last year. Any recommendations for a state park in VA somewhere?