icloud old email address error

I have changed my Apple ID email address to a new e-mail address and with the new email address I can access to my apple ID without any error.

But the icloud app still is registered on my old email address and wont change. When I try to sign out from the icloud it asks me a password but with the old email address. No matter if I write the correct password it wont sign out because it is still shows my old email address. My old email address doesn't exist any more because I deleted it. I thought that when I change my email address of my apple ID, it should update all my account information.

But it left out icloud and its giving me errors all the time. Need help to solve this problem!!!