The reason that you're experiencing difficultly with e-filing a New York state tax return is almost certainly due to the fact that the TurboTax software won't allow that to happen, when the user has first selected (or was forced into) the paper file option for their federal tax return. There is no way to "fix" this, or to change the software program to accommodate a New York state e-file.
The underlying reasons for this, however, are that the New York Department of Taxation and Finance needs a taxpayer's Form 1040 (federal return) in order to process the state return. Where there exists no e-file version of the TurboTax user's federal tax return, it is not possible for the software to create one just for a New York e-file effort. Also, in an effort to combat tax return identity fraud (theft), in cooperation with the state revenue authorities, it is not possible in TurboTax to e-file just a state tax return alone, without an accompanying e-file federal tax return.
Thus, despite New York's mandate for e-filing, you are left with a couple of reasonable choices.
Probably the easiest thing to do, however, is to simply ignore the New York e-file mandate (for the very good reason that you are unable to comply in TurboTax, having already paper filed a federal tax return), and print and mail your state return to New York, along with signed copies of Pages 1 and 2 of your Form 1040. Here is a link to the valid, official, New York mailing addresses that you can use for this paper filing purpose: