The New York e-file mandate only applies when the three conditions below are met -
You're required to electronically file your return if you meet all three of the following conditions:
you use software to prepare your own personal income tax return;
your software supports the electronic filing of your return; and
you have broadband Internet access
So, if for some reason you have a situation with your TurboTax return that causes you to need to mail your return, then that means the software does not support the electronic filing of your return and you are not subject to the e-file mandate. You can mail your return without it being rejected. Also, NY removed any taxpayer penalties for not e-filing several years ago.