Voter Registration form defines domicile and penalty for violation
If you can't prove residency by documents (normally a NH license showing your address) then you can swear out a domicile affidavit. Fibbing on your affidavit is a bad idea, with penalties up to a year in jail and a $5000 fine. The jail is unlikely, but the fine isn't.
RSA 654:12 Voter Qualifications
RSA 654:12 describes the procedure followed for all registrations where a photo ID is not presented.
Then acording to you the guy on Rush was lying.