@ Αnnie_M: Ι've just signed up, thanks! I'm afraid you are right, I need to contact FB support, but how can I do that? I've search the internet on how to contact them and it seems it's impossible if you are logged out! I'd be grateful if you could suggest a way to do that.
@
kataran: I've tried to log in on the net but still, FB requires that I enter the 6-digit code. When I click "need another way to authenticate" it prompts me to go to FB app and find the code generator. But I am logged out from there too! And there is no other suggestion nor a link to contact FB support if you keep facing an issue.
What I've learned from this experience is that a) if you have activated the two-factor authentication and you get logged out from FB on all devices, you are basically doomed.

And b) contacting FB is so hard.