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You may want to start this process by logging out of your AppleID or at least turn off find my iPhone/iPad/Mac. 
Failing to do so can lead to problems: in my case (did this a week ago) one of my devices (a mac with latest OS) failed to automatically update the icloud account to the new AppleID, and in order to sign out and do it myself, I was asked to provide the password to the AppleID email that was nonexistent already at that point in time.  One does not simply log out a turn off findmydevice with a single tap. And that makes sense.
I ended up having to switch AppleID back to what it used to be, log off, confirm password to turn off findmymac, change it again, login again. All that pain with two factor auth... oh man, I wasted about an hour.

Long story short: SIGN OUT before you change your AppleID or spend the fool's hour.

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