I have only foreign earned income (no W2!), cannot e-file my federal return and my NY state tax return must be e-filed. I cannot submit my returns via TurboTax online.

You will have to print and mail both returns.  Returns with no taxable income at the Federal level cannot be e-filed.  New York only has to be e-filed if it's possible to do that.  Your state return cannot be e-filed because you have no e-fileable Federal return.  you can print and file your state return without penalty.