At Assateague, you can camp with the wild horses. there are two campgrounds - one is the national park, one is the MD state park. the MD state park has bathrooms w/ hot water. the horses literally walk through the campsites and you're just a sand dune away from the ocean. It's frightening and beautiful at the same time. Like no other camping experience.
Cape Henlopen is another beach camping place near-ish to Rehoboth Beach. Small sites, sort of a hike to the beach, but it's still on sand, and you have functional bathrooms nearby.
Both are around 3-4 hours away.
Cunningham Falls is definitely one that gets mentioned a lot. Near that is Greenbriar State Park, which has campgrounds in a forested area, walking distance to a nice lake with a swim area and "beach."