If you can still sign into your account on another device (such as a computer also with you on the trip) you can investigate adding more ways to get a code such as multiple telephone numbers where you can receive a text or a phone call. This is from the section "About trusted phone numbers and trusted devices" in the article: Two-factor authentication for Apple ID - Apple Support
To see, add, or change your trusted phone numbers use one of these methods:
● On your iPhone, iPad, or iPod touch: Go to Settings > your name > Password & Security. Next to Trusted Phone Number, tap Edit.
● On a Mac computer choose Apple menu > System Settings (or System Preferences), then click your name (or Apple ID). Click Password & Security, then add or remove a trusted phone number.
● Using a web browser: Go to the Account Security section of https://appleid.apple.com .
Similar information for the iPhone can be found here --> Manage two-factor authentication for your Apple ID from iPhone - Apple Support
If you had set up an Account Recovery Contact ( Set up an account recovery contact - Apple Support ) you can also get a code that way.
If you can’t sign in, access a trusted device, reset your password, or receive verification codes, you can request account recovery to regain access to your account. This article explains how to recover your Apple ID after a waiting period - How to use account recovery when you can’t reset your Apple ID password - Apple Support
I agree that this is an annoying scenario. The best thing would be to set up a trusted number or account recovery contact with a trusted person who could then tell you the verification code while on the phone with them when setting up your new phone.
