There probably will be freezing weather that time of year. One of the interesting things about New England is the unpredictable weather. It could be freezing one day and in the eighties the next. The foliage is almost as unpredictable, but if you catch it at all, it can be fabulous!
As far as parks go, I haven't done much camping in NH, but do recall a place I stayed in Epsom 16 years ago called Circle 9 Ranch. Nice place with its own dance hall and a band on Saturday night.
Since many parks close around Colombus Day, make sure you call ahead to confirm that they're still open.