Do I have to mail my state return if I've already mailed my federal return or is there an option to e-file my state return even after I've printed it out?

Yes, you will need to mail the state return as TurboTax software will not allow a state return to be e-filed if the Federal return was not also e-filed. 

The New York e-file mandate only applies if the software you are using supports the electronic filing of your return and, in your case, it does not.

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

https://www.tax.ny.gov/pit/efile/efile_mandate_for_individuals.htm