The key is to do the states that you are not a resident of first. Then you end in the state where you ended up in at the end of 2021. Different states have different rule about what they tax but they all give credit for taxes paid in other states. It is possible that some of the income you made will get taxed more than once but as a part year resident of these states you should be able to separate out what income was earned in each location so that you aren't paying taxes on the same income over and over.
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