Not that Mt. Rushore, Yellowstone, the Petrified Forest, Grand Canyon or Brice or Zion Canyons are ON the east coast or anything . . . .
Personally, I'm no big fan of trying to hot-foot it clear across the country. I've done that, and it makes for a stressful trip. I AM a big fan of the 2-2-2 rule: drive 200 miles, arrive before 2 PM so you can get the best campsite, and stay for 2 days so you can see a few sights.
But with the caveat that you will at best have 14-17 days to spend going up the East Coast, I'll throw in a few suggestions: Charleston, SC (stay at the Charles City County Park campground -- it's the best!) offers a lot to see and do. There are a LOT of beaches in North Carolina that are absolutely beautiful. There are, however, NO beaches in Virginia Beach, but nearby Colonial Williamsburg is fabulous. The drive up the Blue Ridge Parkway and Skyline Drive is beautiful if you prefer mountains to beaches, and there are lots of low cost (no hookups) campgrounds all along them.
In DC, be sure to visit the Arboretum along with any of the museums and monuments along the Mall. And you are wrong about spending a week in DC. You could spend two or three.
