Try Parvin state park?
https://www.state.nj.us/dep/parksandforests/parks/parvin.html
They even have hunting but it looks like it's zone 21 not 20. But probably closest least expensive option to zone 20 unless you stay at a truck stop.
Campsites: 56 tent and trailer sites with fire rings, picnic tables, lantern hooks and a playground available. Six people and two vehicles per site. Flush toilets, showers, and laundry facilities are within walking distance. Facilities available for people with disabilities. Trailer sanitary station. Located on the south shore of Parvin Lake. Open April 1 to October 31. Fee: NJ Residents: $20 per night; Non Residents: $25 per night.
Pet Friendly Campsites