Changed Apple ID email, new email not showing on Mac so can’t log in

I changed my Apple ID email address. Now my MacBook keeps asking me to enter my password to continue using iCloud, but it’s showing my old email address so its saying that my details are incorrect. It doesn’t let me type in my new email address. If I select to log out it brings up a message saying things will be deleted so I don’t want to so this. It feels glitchy so I’m worried I’ll lose my stuff.

How do I log in to my iCloud account on my laptop without deleting anything?