Welcome to New Hampshire.
The first thing I would recommend doing is getting your NH motor vehicle operator license. They don't give you a laminated plastic license at the DMV, but instead provide a printed on computer-paper temporary license that is good for a month.
They will mail you the laminated "permanent" license within the month time-frame, to your NH resident address.
Print, and then fill out this form:
How it worked for me in Manchester was I went down to the police station, submitted the form after verifying with the Officer who handled my app that it was complete. Make sure it is completed, as it can delay things for you if it is not. There is no application fee when you submit the application. All I had to do was provide my temporary NH driver license to prove that I was the individual in my pistol license application.
I provided my phone number on the app, and they called me back at day nine to come back to the station to pick up my license. At this point you will owe them the $10 fee, and they will give you a receipt for it along with your pistol license.
A trust is always the best way to go with NFA items to avoid cleo issues.
I use "the firing line" in Manchester as my indoor range.
Good luck to you, and again welcome to NH.