It is not giving me another option, just asking for the 8-digit code after I enter the email and password. When I click on try another method, it says you didn't give enough information to verify that this account belongs to you.

Even if you do that same thing several times? (For me it eventually changes)

Let me try this, should I keep on doing this again and again or after a gap of few hours/days?

Again and again, just 4 or 5 times.