If you come down the eastern shore, there are commercial camps up toward Delaware and there is a VA state park at Kiptopeke in Cape Charles, just a few minutes north of the Bay Bridge that takes you into Va Bch. It's a nice campground, great if you have a boat as it is right on the bay. A bit north of that is Cherrystone, another commercial park, quite large and lots of amenities. I've stayed at both, and if you have young kids they will probably appreciate the bigger park, Cherrystone. Kiptopeke is better for folks wanting a quieter stay.
Weather should be nice by April, chilly nights but cool to warming days.