Changed Apple ID not updating on phone or Macbook

I changed my Apple ID. It was changed successfully - I got the email notification, and I can see this when I login on a browser, but on my phone and Macbook it still shows the old Apple ID.

My phone is prompting me to sign in again, but won't accept any password. It says 'verification failed' - because it still shows me has logged in with the old Apple ID. And there is no option to change it.

This is the same account, I just changed the Apple ID (email address) associated with it.

Any idea how to get my phone and Macbook to show the new Apple ID, without needing to log out and back in again?