Account verification when overseas

I live in the United States and was vacationing in Rome, Italy. I was using the hotel computer and tried to sign into my Microsoft account to check my email. Microsoft told me they didn’t recognize the computer and would not let me sign in without verification. They wanted to TEXT or EMAIL me a pass code number that would let me sign in. I did not have a phone, that they could text to so, that option was out. My alternate email was my Gmail email which I had given Microsoft. Unfortunately Google was telling me the same thing. They could not recognize the computer so I could not sign into that account either, to get the password. I had used my Microsoft email as the alternate with Gmail. I was locked out of both accounts. I cannot have this happen when I am overseas. Is there a way around this in the security setup that I am not seeing? Can I have a code that I can use anytime without having to use the alternate email or Text? Or can I disable this security feature with one or both of these email services while on a trip so the two companies don’t try to verify me?