Q: Can't sign into iCloud on my iPhone 5.

I continue to receive the same message pop-up multiple times a day on my iPhone.  It wants to sign into iCloud using my old Apple ID (an email address that doesn't exist anymore).  I have read and tried many different suggestions to which none have resolved my problem.  Does anyone have any methods that work?  Just today I tried to reset my password for the old Apple ID and for some strange reason, the birthdate that I entered they say is wrong, so I can go that route.  And then in order to delete the id from my phone, I have to have the password.  Which I thought I knew what it was, but apparently I don't (plus it says the Apple ID or password is incorrect).  I am guessing it realizes that Apple ID isn't valid anymore?  So, I can't sign in and I can't sign out AND I keep getting a pop up that it wants me to sign in.  Please help!  How do I change my Apple ID for accessing the iCloud?  Thanks!