Can't Change Apple ID On iPad

I recently changed my Apple ID. I changed it in iCloud, iTunes account and, followed instructions to change it on my iPhone but I can't seem to change it on my iPad. To sign out of my account, I first need to enter the password for the old ID but, the password has been changed too. It won't accept either.

To get around this, the advice is to go to my iCloud account and temporarily change it back. The problem with that is, iCloud sends a verification number to that email address and that email account no longer exists (hence the reason for changing it in the first place). That leaves me stuck. I can't change my ID to the current details and I can't go back to the old one.

Can anyone help?