Sign out of iCloud account with password that is no longer valid
I changed the email and password on my Apple account, but my iPad is still signed on to iCloud under my old email. I cannot sign out of iCloud on my iPad without turning off Find My iPad, but I cannot turn off Find My iPad unless I enter the password for the old email ID, which is no longer valid. When I enter the password that I used with that old email, it says it's not valid, but I know it is the correct password for that old email. When I try to sign into the Apple account using that old email. apple.com tells me that user ID does not exist. HELP!
iPad, iOS 8.4.1