This is incorrect, it comes down to whether the town/state bothers to expire it when you send them the moving notice. There are also situations such as having a business address still in MA that would allow you to simply use that address for your "Resident" LTC.
Either way it doesn't matter because the ATF has said if you are a part time resident, then during the time you live in a state, you are a resident of that state for firearms purchasing purposes. MA mostly cares about making sure they still tax your income if you are a part time resident.