I have to mail my Federal which means I also have to mail my state, but NY is currently ONLY accepting efiled state taxes? How can I get around this and mail my federal but efile my state?

You can mail your NY return.  It is not possible to e-file a state return unless you e-filed the federal return.

Here is the address for a mailed NY return.

https://www.tax.ny.gov/pit/file/return_assembly_mail.htm

Why do you have to mail federal?  Maybe we can help you fix it so you can efile Federal.

I accidentally listed myself as non-filing for 2019 so I have to send in an amended return for Federal 

I have the address, but when I printed my return to mail it from Turbotax there was a page included that says:

"New York State requires this income tax return to be filed electronically.

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.

Besides being the law, e-filing has many advantages, etc etc etc". 

Again---if you cannot e-file it, NY will accept a mailed return.  

NY requires Efiling *IF* your software supports it.  If you can't efile federal then Turbo Tax doesn't support Efiling state.  So you can mail it.

Okay, do I have to include a written declaration as to why I can't e-file? 

Would it be safer just to print the tax forms and enter it by hand so it doesn't look like something printed from software? 

No and No.  Don't worry about it.

Stop stressing about it.  You are far from being the only person who has had to mail a NY tax return.