How do I change my Apple ID on my iMac. It is showing an old ID in System Preferences/Internet Accounts and System Preferences/iCloud

My iPhone, iPad and iMac Safari access to the Apple Store and iCloud are fine and accept my new Apple ID. Mail constantly prompts me for a password with a box that shows my old Apple ID. I'm guessing that somewhere on the iMac I can change or delete the old Apple ID? It's not in the Keychain.
iMac OS X 10.10
Mail 8

Thanks Eric. If I try, I get a box with the statement:
Your iCloud account couldn't be removed at this time.
Mail was unable to delete [email protected] (It just shows this as normal black type without the email envelope) Relaunch Mail then sign our of iCloud again.
It shows an OK box.
The Apple ID shown is of course, the old one, that I am trying to change on the iMac. On my iPhone, iPad, the Apple Store and iCloud via safari, only my new Apple ID is shown.
I tried to insert an image of this scenario but I couldn't insert the chosen 138KB tiff image.

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