We have spent several Christmas to New Years' at Hunting Island SP. It's $30/night, right on the ocean, water/electric only, but has a couple of dump stations. Unless the weather is really cold, it is usually pretty full. Lots of locals come down for the holidays.
The weather can be iffy, but it is NORMALLY, 60s during the day, maybe a little warmer and then gets down in the mid 40's at night.
If you are in one of the spots in the back, it is usually warmer since you don't have the breeze directly off the ocean.
Yes, I would make a reservation. Sometimes, they close some of the loops.