Yes, you need to file a tax return in each state where you physically worked as a part year resident.
As far as filing jointly, state law may dictate that and it may be easiest to file separately. Most states allow you to file jointly even if both of you do not have the same residency status, however some state require that you file separately if your residency status is not the same. Click here for specific details by selecting each state. You will want to use the method that results in the lowest overall tax.
If you find that filing jointly is allowed for all states you must file, then you will be allowed a credit for taxes paid to other state for the amount of tax liability for each other state, on the DC return filed jointly with your husband. In your case it would be for MA and NY. A credit for taxes paid to another state is the lower of the tax paid (liability) to the other state or the tax that would have been charged on the resident state (DC) return.
If you file separately, then there is no credit on either DC return.
See the example below where you would enter a credit on the DC return filed jointly.
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