My husband just got a new phone and can no longer log in. He has had two other devices that his watch synced to, but he is now not even getting an email to change his password. He doesn't want to lose his data.

I assume you have your own account. You can check for the existence of his registered email address with Garmin.

On your phone, logged into the app as you, go to More > Connections.

Tap the search in the top right corner and enter in all his known email addresses to see what shows up.

At least that would narrow down which email address he has been using. Then he should try reset the password and make sure the Garmin emails are not landing in his junk folder.