In NH's White mountains you'll find a whole string of national forest CG's, probably 20 or 30 of them, take your pick. The Kancamagus Hwy (112) and 93 are worth doing.
State parks in Ma., Myles Standish (not on the beach) is just outside Plymouth and Salisbury Beach is north of Boston, it would be a good spot to check out Rockport, Gloucester and Newburyport.
In Ct., maybe Rocky Neck or Hammonasett SP if you want to go I95 along the shore, both have beaches. I prefer the Hammonasett beaches, but that's just me.
Fisherman's Memorial SP in RI is a good one too, also along the shore. Light house, a couple decent beaches nearby, Narragansett Peir, Newport is close by too.