In NH, Danforth Bay Campground is about 45 minute east of most major White Mountain attractions. DB gets great reviews and has lots of amenities. Well run campground, overall clean. large wooded sites for the most part.
Also might want to consider Fransted family campground or Lincoln Woodstock KOA both get great reviews and are more centrally located to all White mountain attractions
Scenic Kancamagus highway is a must drive, but you will be VERY luckie if you see a moose as they are in decline.
In VT check out
Quechee Pine KOA centrally located in vt, good reviews,close in interstate.
Hope this helps
Mike