I believe you may be mistaken. I just read through the NH regulations, and find no restrictions at all on what shotgun you can use. All I can find is that a bow must have a 30 pound draw. In the prohibited section, it states "It is unlawful to take a turkey with a rifle, air rifle, dogs, electronic calling devices, live decoys, or while it is perched in a tree."
I then dug into the NH general regulations, and found nothing there. I did find the following quote in the restricted section that has to be the single most bizarre statement I have ever read in a hunting regulations book.
"Negligently shoot, wound, or kill a human being while hunting, or abandon a wounded or killed human being"
That has to be written???
Anyway, in NH it appears you can use any smooth bore gun, with any size shot, and presumably you could shoot a single ball. I'm not seeing anything that says you can't hunt them with handguns either.