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!