I'm thinking about moving to southern New Hampshire.  I did some searches but couldn't find any place for non NH or MA retired officers to do annual quals.  My department won't do it and the local training center said that they will only do it for retirees that currently live in NYS.  Any suggestions?

Call the Sheriff office in the county you will be living in and ask them if they have open range dates when retired police officers can firearms qual.  In Iowa the SO issues carry permits and also issues the LEOSA 50 state carry permit and no charge for the qual or the LEOSA permit.   Note retired police are still not exempt from Gun Free School Zones and federal buildings and some states laws on mag capacity apply to QRLEO.

https://gunsnh.com/classes/NH-firearms-certification

They do MA residents or officers retired from MA.  

I am a MA certified LEOSA instructor for those who live in MA or retired from a MA agency. I do this in NH at Londonderry Fish & Game Club on request.

I retired from MA and now live in SE NH. I have found it tough to find LEOSA certification in NH. Probably the biggest reason is that a NH certified LEOSA instructor has to be an active LEO in NH. The day an officer retires, he loses his POST certification and can no longer qualify anyone.