It can be done--here is the U.S. Postal Regulations on the subject:
https://pe.usps.com/text/pub52/pub52c4_009.htm
One way if you want to ship via USPS, then coordinate through a local FFL that ships it (only way to do handguns via USPS btw) which contacts a receiving FFL in the state you want. Or, you can ship longarms via USPS to an FFL dealer in the state where you want to pick it up. And, reverse the process before returning home.
https://www.atf.gov/firearms/qa/may-nonlicensee-ship-firearm-through-us-postal-service
You can mail it to yourself c/o someone in the state--but they cannot open the package--which would constitute an illegal inter-state transfer I guess. If you note below, you must also be able to do so under state and/or local laws which requires research.
From hotlink above: USPS regulations
"432.3 Rifles and Shotguns
"Except under 431.2, unloaded rifles and shotguns are mailable. Mailers must comply with the rules and regulations under 27 CFR, Part 478, as well as state and local laws. The mailer may be required by the USPS to establish, by opening the parcel or by written certification, that the rifle or shotgun is unloaded and not ineligible for mailing. The following conditions also apply:
- Subject to state, territory, or district regulations, rifles and shotguns may be mailed without restriction when intended for delivery within the same state of mailing. These items must:
- Bear a “Return Service Requested” endorsement.
- Be mailed using a class of mail, product, or Extra Service that provides tracking and signature capture at delivery.
- A rifle or shotgun owned by a non-FFL may be mailed outside the owner‘s state of residence by the owner to himself or herself, in care of another person in the other state where he or she intends to hunt or engage in any other lawful activity. These mailpieces must:
- Be addressed to the owner.
- Include the “in the care of” endorsement immediately preceding the name of the applicable temporary custodian.
- Be opened by the rifle or shotgun owner only.
- Be mailed using a class of mail, product, or Extra Service that provides tracking and signature capture at delivery."
Otherwise, you can ship as a non licensee via UPS or Federal Express to an FFL dealer in NH and reverse the process before heading home.
https://www.atf.gov/firearms/qa/may-nonlicensee-ship-firearm-common-or-contract-carrier
It might be a bit expensive and cumbersome but how much is your liberty and peace of mind worth.