You're right, Dogtown, my statement wasn't complete enough for my intent.
A handgun size smoothbore is always a "sawed off shotgun" in the legal sense, meaning there is some category of the NFA it falls under.
Sorry, 9ballbilly, I don't know the specifics of the applications. Going to the ATF websight would probably be a good start. I do know that ATF approval for machineguns includes sign off from your local Top LEO as part of it. I assume it might be the same for AOW category, but don't know for sure.
I grew up in NY, but haven't lived there in decades, so I don't know where to start, there, either. But I'm sure you could find some place on the web to begin.
All else being equal (and it almost never is) bigger bullets tend to work better.