It depends somewhat how close to DC you want to be. If you're within reasonable public transit range, you'll have a hard time getting everything.
Shaded and I think some (relatively few) pull-through sites, trees, etc., but no hookups: Greenbelt NP, maybe a 45 minute walk/hike to the closest Metro station. When I was there (Thanksgiving week) it was very quiet, in fact nearly abandoned. The cost is very reasonable.
Some pull-throughs, full hookups, nice landscaping and good management, but not quite so shady or secluded sites: Cherry Hill RV park, with bus or shuttle service to the (not too distant) Metro station right at hand. They consistently get very good marks for catering to tourists and helping with getting around etc. This is probably the closest to what you're looking for.
Somewhat in-between, I think: Lake Fairfax county park, in Reston, VA, a not too long walk to the Metro station. They have electric only sites, I think none pull-through.
I think these are the three campgrounds closest to metro stations around DC. If you're willing to drive some and/or take commuter rail, there are some more options...that I'm not at all familiar with myself.
