You can have only one domicile, but you can be a resident of more than one state. It appears that you meet the definition of a resident for both New York and Connecticut. That means that you would have to file resident tax returns in both states. You should be able to claim credit on one of the state tax returns for tax paid to the other state, but you only get the credit from one state, not both.
This is obviously an unusual situation. I suggest that you consult a local tax professional for detailed guidance.
