In general, you must report all your worldwide income and pay tax on it to the state where you have your permanent home address, also known as your domicile. Your domicile is your main address, where you live most of the time. You can only have one domicile at any time, even if you move between several residences. Your domicile is the home where you have the most loyalty, the most connections. There is no single factor that determines where your domicile is, you have to look at all factors. Things like, where do you spend most of your time, where do you work, where do you have the most friends, the most social connections, your family doctor, your attorney, your significant social arrangements like churches and clubs, your vehicle and voter registration, and so on. In order to change your domicile, you must both establish a new domicile, and take active steps to abandon your previous domicile. Then, if you were living temporarily in another state, you will generally owe income tax to that state for income earned while living or working in that state as well. You would file a non-resident return for the second state and a resident return for your domicile state. Your domicile state will give you a credit for taxes that you paid in the second state which reduces the burden of double taxation.
So as long as you are domiciled in New York, you will owe New York state income taxes, no matter where your employer is located.
Once you have moved out of New York State (and this requires both that you establish a new domicile and that you abandon your New York domicile, by doing such things as selling your home or abandoning your lease, changing your voter and car registration, getting a new doctor and a new church, and so on) then you will owe state income tax in your new state for all your worldwide income earned from the day that you changed residences.
In TurboTax you would file your tax returns as a part year resident for New York and a part year resident for the second state, and you would allocate your income to each state based on when you earned it.
