Already e-filed federal return with H&R Block - Only need to e-file NY State using TurboTax, but TT is throwing up an error about federal taxes needing to be e-filed first?

That is correct.  You cannot e-filed a state return from TurboTax unless you have e-filed a federal return.  TT cannot be used to prepare "stand-alone" state returns.  Why didn't you use the same program for your state returns that you already used for your federal?

And despite whatever H & R Block said---it is possible to mail a NY return.https://www.tax.ny.gov/pit/file/return_assembly_mail.htm

As I said in my OP, H&R Block stated it couldn't e-file a part year resident NY state return.  And NY State's return states:

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New York State law requires income tax returns prepared using software to be e-filed. Because this New York State tax return was prepared using software, you MUST file it electronically.

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So I bought TT, and re-prepared my federal as well as NY state (I didn't "stand-along prepare" just the state). But TT won't let me file just [NY state], which is dumb.

too many state returns filed by fraudsters, so that's the reason for this restriction.