There's a lot to factor in here. If you're driving by yourself, realistically you can do about 12 hours a day. So 4 days, that means 3 nights in a hotel (assuming you can go straight to your apartment or wherever the day you arrive in NJ). Plus food, and about 100 gallons of gas. Yeah, you could just sleep in the car at rest areas, but having a bed and a shower makes a huge difference in your moral and back pain.

If you ship the car, you can also fill up the car with some of your stuff so that saves a bit on moving costs. You'll be without a car for probably a couple of weeks so be sure to factor in the cost of Uber or other public transportation. Then also the flight cost.

I helped a friend move from NY to AZ awhile back. We took 8 hour shifts each of driving, and stopped for the night in Indiana and western Kansas. This country is so huge, and most of the driving is incredibly boring. I really appreciated having a 5 hour flight home versus 3 days in a car. I was so sore afterwards.