NJ campgrounds

The Depot is the closest to Cape May beaches( maybe 10 blocks? ) and is an ok campground to use in my opinion.

Liberty Harbor is what it is, but I'd say it's the best spot for visiting NYC. Anything else is a fairly long trip either driving or by train. Liberty Harbor is a short, safe walk to the Path train.

Every September,we're in Holly Shores which is about 4 miles north of Cape May and a short hop to Wildwood. You might find Big Timber Lake to be good also, altho it's probably 16 or 18 miles from Cape May.

Holly Shores

Big Timber Lake

Liberty Harbor is just a parking lot with hookups, but it is a good location for what you want to do - and there are almost no other options in that area!
Suggest you check with www.rvparkreviews.com for info from campers on CGs in Cape May area. Also be aware that there is Cape May, then Cape May Court House, two different towns! Most of the CGs in this area are filled with seasonal campsites, and many were created when rigs were much smaller. While most have mostly level sites, they are almost always very, very close together. I would avoid the KOA in the Cape May area (used to be Lake Laurie CG). They have created a newer section for larger rigs, but their ownership seems reluctant to employ enough staff to keep the place even reasonably clean. If you do plan to go there in July, I would seriously consider making reservations ASAP, particularly if your stay would include days over the weekends. These places are costly, but they fill up fast almost every weekend from April through the end of September.