I have pressed File By Mail, then Continue. It does not take me to an instruction screen, it takes me back to the screen where you choose how to file... again.

Have you paid your product fees upfront by credit/debit card?   That's required in order to file by mail.  You cannot choose to pay the fees out of the Federal refund.   See these steps:

https://ttlc.intuit.com/community/printing/help/how-do-i-print-and-mail-my-return-in-turbotax-online/01/26258

I had a similar problem with the e-file option. I was using the downloaded version of TurboTax, not the online version, so there was no payment issue. I had entered banking information in the questionnaire. I had to delete that information in the questionnaire. When I got to the e-file screens, I had to set up new bank account (which was same as the old one) and then verify the information. That worked. It might be the same for filing by mail if you opted to have your refund sent to your bank account.