I FOUND THE SOLUTION!
I had the same exact problem. I had my old apple ID connected to a school email which did not exist anymore. My ipad was prompting me to put the password in for that ID and I could not delete it from my ipad because the "Find My Ipad" was turned on, once again prompting for password. I called apple and they said this Apple no longer exists and is not in there system, which I knew already. After hours on the phone nobody could help me. Try this.
-Go to create a new Apple ID and use the same email address as the old apple ID which is locked in on your ipad
-Create a new password and all that, it will say it send a verification link to your email, in my case my email no longer exists so I could not do this
- I called Apple to explain this problem and I asked if they could verify it for me, which they did after I convinced them it was worth a try. The guy told me that the verification shouldnt matter and should still be able to login but he did it anyway. You can try skipping this step but if it doesn't work then go ahead.
-Go to ICloud and click on your account, the email should be grayed out but it should allow you to change that password, put in the new password which you just created.
-This allowed me to sign in to accept the new terms and conditions but it said "unable to conncet to server" so I just backed out and went to "Delete Account" and it worked!
The whole idea is to re-create your old apple ID into Apple's system and then use that new password. The guy at Apple said tons of people called about this problem and nobody could solve it. I thanked him for helping me and he thanked me for helping him. I hope this helps you!