You can also post an ad in the EE that says "FTF in NH only," which means you will meet "Face to Face" only, so no FFL is required for a person-person sale. You can also mail a lower to another resident of NH without the need of an FFL. In that case, make your EE ad say "FS in NH only."
From the BATF Website:
B9) May a nonlicensee ship a firearm by carrier? [Back]
A nonlicensee may ship a firearm by carrier to a resident of his or her own state or to a licensee in any state. A common or contract carrier must be used to ship a handgun. In addition, Federal law requires that the carrier be notified that the shipment contains a firearm and prohibits common or contract carriers from requiring or causing any label to be placed on any package indicating that it contains a firearm. [18 U. S. C. 922( a)( 2)( A) and 922( e), 27 CFR 178.31]