If you purchased an additional state download, you can download it as many times as you like for up to 3 years from purchase. You should find the NY state download in your "Downloads" section in your TurboTax account. The TurboTax FAQ I'm linking below will instruct you on how to download this purchased software to your new computer.
How do I download the software I purchased at TurboTax.com?
*** I am NOT a tax expert. I am a seasoned TurboTax user, and volunteer to provide assistance to TT users. Nothing I post is to be considered TAX ADVICE; I bear no legal liability for responses.***