Sorry--you cannot use TurboTax to prepare just a state return. Information flows from the federal return to the state return. In order to use TT you would have to purchase the desktop download software to use on a full PC or Mac, re-create an exact duplicate of the federal return and then do the NY return to mail in. Do-able---but tedious and painful.
https://turbotax.intuit.com/personal-taxes/past-years-products/
A 2022 return cannot be e-filed; it has to be filed by mail. Do you have a copy of the rejected NHY return that you can mail in to NY?
Or---try going to the NY tax site to see if you can get the forms from the state to prepare by hand. You will need information from your federal return so make sure you have a copy of your 1040 handy.