I would be thinking about statute of limitations. You might want some professional advice.
It is likely that the statute of limitations starts when you file, so there is no statute of limitations for NY and they can tax you even though it was more than 5 years ago.
The problem will come when you file an amended return in your home state for 2018 to claim the offsetting credit for taxes paid to another state. It might be too late to get that credit refunded to you (but there might be an exception when you have to amend due to a problem from another state).
Also, Turbotax is not supported for 2018 any more, so even if you have the program installed on your own computer with the state module for your home state, you won't be able to install the NY module. So you will either need to prepare your returns by hand or see a professional.
Note:
You report and pay income tax in your home state on all your world-wide income, no matter where you were living or working when it was paid. You owe a New York return for income that was earned or paid while you were living or working in New York--the difference with the 183 day is whether you are considered a resident or non-resident. You are a non-resident for 2018, but that still means you own tax on the portion of income that you earned while physically living or working in NY. But, the tax NY is proposing may not be calculated correctly, since they may not be allocating the right amount of NY income. So you should prepare a 2018 return and allocate your income yourself, based on how many days or weeks you were physically in NY. Then, you can file an amended return in your home state to claim a tax credit for taxes paid to NY. But this may have issues since it is so late, and you may want to check with a professional.
