Yeah, most anywhere you go- you can either get close to the beach or camp next to it. In a few cases, you can feel like you're *on* the beach.
As mentioned, Cape May and the surrounding area have quite a few places that are 10-15 minute drive.
Rehoboth Beach has Delaware Seashore State Park.
Maryland has parks in Ocean City.
Myrtle Beach has beachfront campgrounds.
And on and on.
Sounds like an excellent trip!
Though, Key West really doesn't have beaches - there are bayfront/waterfront campgrounds there, too.