If you want to be in New Jersey and don't want to stay at Liberty Harber, your next best area is Northwest Jersey. Fla-Net is in Netcong, off I-80 at Exit 27. From there it is about an hour to Brooklyn via Staten Island during off-peak hours. There is also a campground in Hope, N.J. off I-80 at exit 12, about another half-hour west. Mahlon Dickerson county park in Morris County is off Rt. 15 and has at least electric hookups if not also water.
If you are coming from the south, Turkey Swamp is a county park in Monmouth County off I-195. It has water and electric. There are also some private campgrounds in that area. Access to Brooklyn is again through Staten Island.
I don't know of any campgrounds worth staying at along I-78.
Except for Liberty Harbor, that's the best you are going to do in New Jersey, an hour to an hour and a half drive from the city.