Does this mean that I should not be paying NY state income tax and only CT income tax?
Correct, if you meet the criteria, then you should not be paying taxes to NY state, only CT.
Is there a way to let the State of NY know this? Yes, file your tax return as a NY non resident and allocate $0 to NY. This will give you a refund of all of the taxes paid to NYS.
You will then need to have your employer change the state of withholding. Since they are probably still withholding for NY, you will need to probably file the same way again for next year.
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