Since NH does not have any state income tax, you are correct, you do not need to do anything with NH. Also, since you worked in MA all year, ALL of your income would be taxable to MA first before NH (if NH had a state income tax)
As for filing MA, yes, you will need to file the MA Part year return. As you walk through your MA Part year return, TurboTax will ask you about the days you lived in MA vs NH and also how much of your income is associated with MA. Since all of your income came from MA, you may still qualify for some credits in their entirety, while others may be reduced.
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