Here in Vt the state parks are open to visitors but gated and all facilities shut down after Oct. 14. With a frost depth of 4 ft., running water is unlikely at any campground. They will have shut the water off by then to avoid frozen pipes.
Last winter there were some hardy folks who posted about camping at Rasberry Crossing (NFS) on the eastern side of the Kancamagus Highway in New Hampshire. Pit toilets and a hand pump. While the Kancamagus is open year round, pulling a trailer over at that time of year might be an adventure. True for any of the passes.
There is a good chance for a little snow and with it sanded/salted roads. You will want to wash the underside of you trailer when you get home.
Finally, Thanksgiving weekend is the last weekend of rifle season so leave your deer costume at home.
Camp in the White Mountains of New Hampshire