For the Adirondacks (and New York state in general), a very handy resource for public campgrounds is
the campadk campsite photo database. I tend to consider the Adirondacks an honorary part of New England, for what it's worth. :C
In Vermont, the state park campgrounds all lack any hookups but offer (with maybe one or two minor exceptions) a dump station/water fill station. Many of the campgrounds date from the CCC days, and so tend to have small sites well-separated by vegetation and such. Larger RVs often only fit in a handful of sites.
Mt Ascutney offers some nice hiking (both relatively strenuous if you hike up and down the mountain, or more sedate if you drive up and merely hike around the summit) and impressive views. The half dozen or so state parks in and around Groton State Forest interconnect with a network of trails that offer good hiking, and hiking that isn't just up or down a mountain. There's a neat little stone shelter on Owl's Head there. Little River state park also has some hiking trails among the ruins of a former settlement (mostly cellar holes). There are other Vermont parks with good hiking and historical points, but those are ones that I've personally been to and enjoyed.
St Johnsbury is an interesting city to spend a little bit of time in; the Fairbanks Museum is a neat old natural history museum, with some other quite eclectic items, and in some ways also a bit of a "museum museum" in that it hasn't changed a whole lot in many years and so shows something of what museums used to look like. The Athenaeum (public library) has a very good little fine art gallery, too.
In New Hampshire, the White Mountains have lots and lots of things to enjoy and places to see. I'd suggest at the least a leisurely drive over the Kancamagus Highway, which has stops for various overlooks and hikes ranging from brief walks to quite decent treks.
Boston, of course, has a long list of historically important sites, but limited parking in the city and even more limited camping.
