Some ideas; NJ native here.
The Jersey Shore is generally pretty built-up residential, but there are two parks that are worth seeing: 1) Sandy Hook, located at the very south of the New York Harbor inlet. This is a Federal National Recreation Area, and 2) Island Beach State Park, on a barrier island about mid-point north-south.
From there, to keep going in the southern direction, either continue to Cape May and do the ferry to Lewes, Delaware, or head back to the NJ Turnpike/295 to cross the Delaware Memorial Bridge. For scenery, I would recommend going down Delaware 1 if not taking the ferry, and do the Chesapeake Bay Bridge/Tunnel. If DC is absolutely a thing, use US301 to US50 and cross the Maryland Bay Bridge.
But it seems driving the coast is the intent here, so I'd say the Chesapeake route would be the thing to do. Follow US17 down into NC, where a detour to the Outer Banks could be done.
I'll leave suggestions on points further south to others familiar with that geography.
