Big Camper here - my favorite hands down is Big Meadow in Shenandoah National Park. Probably a 2 hour drive.
f you don't have to shower, then you can stay closer at Matthews Arm in Shenandoah - maybe 1.5 hours away.
Shenandoah has all of the beautiful overlooks and hiking the Appalachian Trail - they little $2 booklets at the visitor center for short hikes, easy hikes, etc.
You can reserve spots at recreation.gov.
When we are more in a hurry, we stay at Elizabeth's Furnace - which is off the same exit as Shenandoah on Rt. 66 (I think its exit 6?) - but it's only 10 minutes from the exit rather than 30+. It has trails too in the Massanutten Range that are interesting. No running water faucets in the bathrooms or showers though - only pump water that you cant drink, so bring your own water. This place is popular, so I would try to get there by noon on a friday or earlier - or reserve a bigger sized group camp spot on recreation.gov (you cant reserve regular size spots).
http://www.fs.usda.gov/recarea/gwj/recreation/camping-cabins/recarea/?recid=73687&actid=29
If you are really trying to stay local, and want to camp in style with really nice showers (big meadows has showers though), I highly recommend Oak Ridge Campground in Triangle VA. It was pretty dead last time I went there - which is fantastic (ahh peace and quiet!). Here is the link:
http://www.nps.gov/prwi/planyourvisit/oakridge.htm