Can I file my federal and state tax return by mail in New York even though it says New York state returns must be e-filed?

Yes, you can mail returns in NY state. 

They highly encourage e-filing, but some people just do not have that option.  It took a little digging but I did find the information of where to mail it in NY, here's the link:  https://www.tax.ny.gov/pit/file/return_assembly_mail.htm. 

The IRS was a bit easier to find: https://www.irs.gov/uac/new-york-where-to-file-addresses-for-taxpayers-and-tax-professionals.