I found the best way off the Island is leaving at the crack of dawn and using the GWB for southern destinations. The BQE is horrible! I've had entire contents of my food pantry empty on the floor, including the three pound BJ's coffee can contents. Drawers have opened, just a crappy road.
I have a seasonal at the Copake KOA and leave from there on our trips, since the FW is there. A good kick off point for you would be Newburgh KOA or Jellystone in Gardiner. Of course I would not use these as a start off point for a trip going south. All the campgrounds I mentioned are good for a few days of chillaxin' as well.
Edit: just realized you were looking for PA & NJ campgrounds