Yes, if you changed residence, you will have to file a Part-Year Mass return and report the income you received when you lived in Mass.
As a part-year resident, you're generally taxed on all your income while you're a resident of Massachusetts, received from sources inside or outside of Massachusetts. Massachusetts gross income is income earned from all sources while a Massachusetts resident.
Do I have to file a part year NH State tax form too just for the dividents or interest? Is that divided into MA and NH or can I just report on the MA state tax form?
If you met the NH income threshold (see below) for interest & dividends, you would owe NH tax on the dividends and interest you earned while a resident of NH (a Part-Year resident return). That income would not be taxable by MA, but you would have to allocate the rest of your 2020 intent and dividend income to MA.
There is a threshold for the NH tax on dividends and interest. From the NH tax website: The state only taxes interest and dividends at 5% on residents and fiduciaries whose gross interest and dividends income, from all sources, exceeds $2,400 annually ($4800 for joint filers).