I've led many school groups to Wash. D.C. as a tour guide, it's a great place to take kids. The campground I've read a lot about is Cherry Hill.
Cherry Hill Park is the Closest Campground to Washington, DC :|: Cherry Hill Park
One of the key things is it's got access to public transportation. You do NOT want to deal with parking in the Mall/Museum area. All the Smithsonian Museums are free, also the National Archives, and all the public memorials. Get a good paper map of the Mall area and plot out each day's key points. I'd also recommend the National Cathedral and the National Zoo, both free.
On your way to Washington, you might want to check out Smoky Mountains National Park, Blue Ridge Parkway, etc. Many routes and sights to choose from on the route from TX to Washington. A few hours beyond Wash. is Ocean City, MD. A true eastern seashore type experience with beach, boardwalks, and more. There are campgrounds in the vicinity, Assateague State Park (wild horses on the beach) is just to the south. To get to Ocean City, you'll travel through Annapolis (Naval Academy) and over the Chesapeake Bay. The Naval Academy is great to explore on foot and many things to see.