Hey,
New Hampshirite here, lol.
Let me give you a a couple places to look at. You definataly want to stay in the Lakes Region. Our favorite place to stay in the state is Danforth Bay. It's on Lake Ossippee, in Freedom. Great campground with tons to do. Located about 30 to 45 minutes from North Conway too. So not too far to get up there and do the things they have to offer. Here's the link: https://www.danforthbay.com/
If you not looking for bells and whistles in the campground, here's one that we go to often. It's called Goose Hollow. Quiet, friendly staff. It's about 3 miles off 93, at the base of Waterville Valley. Very senic location, quiter place, small pool, on the river that runs down from Waterville. It puts you right at the southern part of the Mountain region. 25 minutes to Plymouth (good food/shopping places, Walmart, Groceries, etc) and 30 minutes to exit 32 (the Kancamangus Highway which runs over to Conway, the most senic drive in the state). Here's the websit for them: http://www.nhcampgrounds.com/sitemap/
I can give you more places if these don't work. Not sure how long you are staying, but it would cool to spit your time between them. Two different camping experiences.....