I have done quite a bit of car camping in NH. I love the NH stateparks, but alot of them do not allow dogs, and what is camping without the pooch? Bearbrook Park allows dogs and went there a few weeks ago. Great hiking trails, nice facilities, good bicycling, nice ponds, and people shut up at a decent hour.
I have been to a few Nat. forest campgounds, they are nice, and allow dogs, I liked Wildwood off 118. Then there is the Triple I Road in Thorton, NH. Buy a permit and pull over along the road at a nice spot. No facilities, but it is nice.
I have stayed in a few private owned places West of I-93, pretty good, most have hookups, some are better priced than others. There are 2 nice ones off 25 in the Rumney and Wentworth area that are nice and a few private spots. Would be nice for group to get a block of spots. There was a great place in Haverhill on 25 that we used to go to for a few years. Each spot was over 50 yards apart and most had views of the olivarian valley that would take your breath away. Unfortunately the family that owned it had an internally feud and shut down. the price there was like 15 dollars a night with a load af fire wood that would last 2 nights of good campfires and cooking, miss that place.