Yes, you can file a Part Year Resident State Return for both Massachusetts and New Hampshire. You will give the dates you lived in each state in the State Interview, as you suggested.
For your unearned income, like Social Security, bank interest, or retirement that you receive monthly all year, you can allocate a % to each state, based on the % of the year you lived there.
Here's more detailed info on Allocating Income for a Part-Year Resident.
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