I was stationed at Fort Devens, Mass. in 1983-1984 and spent every free moment I had exploring the New England area. I visited several of the state parks and every one of them was absolutely wonderful, and beautiful. I did not have a camper then, and did not camp in any of the parks, but did enjoy swimming and hiking.
Doing a very quick search on Vermont, Maine, Mass, New Hampshire, and Connecticut, there is no shortage of State Parks. Just do a quick Google Search (Main Camping, Vermont Camping, etc), and there will be links to Reserve America as well as links to the official state DNR camping web site. Not to mention, a multitude of private campgrounds that will come up.
I'm not so sure about the time frame they are open though. Most state parks are open all year long. And you can camp all year through. They just turn the water off after a certain date. For this, you'd need to check with each campground for their dates.
Fall in the New England area is marvelous! Dress warm. It does get chilly.