Jersey Shore campgrounds are not on the beach, usually a few miles away. They do get crowded and can be pricey. We usually hit Holly Shores in Cape May in mid-September.
If you want to be right on the beach, I'd suggest Delaware Seashore State Park. The campground is on the bay side, you do need to cross the highway for the ocean side. Cape Henlopen State Park is also good, but it's a bit of a distance inside the park from the campground to the beach.
Assateague isn't too much further South, as mentioned. We've spent time in the National Park Service campground there. The wind on the ocean side seems to never stop and can really kick the sand around
Safe travels and, above all, enjoy