Had the same issue. Here is the solution.
Problem: Can't change email address because my account is tie to my FB account, thus my Airbnb email address is the same as my FB email address. Every time I try to change my email, the system opens up a FB window and prompts me to enter my FB passwaord, then nothing happens.
Solution: Disconnect your FB account from Airbnb account. After that, you will be able to change your email address successfully!
Step by step guide:
Click on your Photo icon -> Trust and Verification -> Disconnect Facebook
Click on your Photo icon -> Edit Profile -> Enter your new email address and click save -> the system will prompts you to enter your Airbnb password (likely your FB password) -> DONE!
Let me know if this help! (or doesn't)