Any sales across state lines have to be done through a FFL dealer.
In the case of a handgun, it must be a FFL in the buyer's home state. In most cases, that involve shipping to a FFL through a contract carrier. In a case like yours, you can bring the handgun to a New Hampshire FFL of your choosing (in essence using yourself as the shipping carrier, rather than UPS or FedEx), and the FFL can then transfer the gun to the buyer.
In the case of a long gun or shotgun, the deal still has to go through a FFL, but it can be in either state. According to Federal law, a buyer can purchase a long gun from a FFL dealer in another state to take to their home state, provided that the rifle or shotgun is legal in both states. So if you were selling a rifle or shotgun instead, you could utilize either a NH or ME FFL for the FTF deal, provided NH or ME law doesn't state otherwise.