Have you found the NJ campground? If it is open year round, and is zoned for long stays (typically zoned as trailer park rather than campground), there may be trailers fitting your budget for sale or rent, so you don't need to tow it.
Aside from the budget, the biggest gap in your wishes is between what your Grand Cherokee is going to be able to tow, and the kind of living space it sounds like you think you need. What you might tow is going to fall into 19-25 foot range, about 150-200 sq ft of living space. Typical fittings at that size will be a bathroom with small shower, tiny kitchen with 6 cu ft absorption fridge, 6 gallon water heater, waste storage for 3 days to a week, one permanent bed (usually a double or short queen) and a small dinette, seating two large people or 3-4 little ones. Open floor space will be an aisle about 2 ft wide, 10 to 16 ft long. Waste tanks will need periodic dumping, and the fridge may not work in cold weather, because it wasn't designed and installed for that use. You will have to work out how to prevent freezing of your water supply.
For your budget, you might be able to get into a mobile home, about 40 ft long and up to 12-14 ft wide, for more like 480-520 sq ft. Plumbing, supply and waste, will be permanently hooked up in non-freezing space. Kitchen appliances will be residential types, at least apartment sizes.
At low cost, there are some FEMA trailers out there, adapted to temporary stationary use, for intallation in mobile home type parks (rather than campground hookups). Appliances are residential, but small. Sizes are 25-30 ft, 200-250 sq ft. Not suitable for RV use without conversion back to RVs, where you can use them will depend on zoning.
Just some things to look into, find out how these things are done in New Jersey.