This solution no longer works. I called in to Apple Support, and had to request TIER TWO support before i was able to get around this problem. Apple has changed the way they address this problem. The work-around that USED TO WORK but NO LONGER WORKS is to temporarily change the apple ID (on a computer) to the old one. If you try that now, you will be told that a verification code will be sent to the OLD ADDRESS. This is not a solution that worked for me, because in my case the old address was no longer in use.
The solution, as of Jan 2 2016 is this:
1. On your device (iPhone, iPad), go to settings, iCloud
2. On the next screen, tap your account near the top. You will be asked for your Apple ID passworrd
3. Enter your CURRENT PASSWORD that you use with your CURRENT APPLE ID (don't worry that the password doesn't match the password as it used to be with your old apple ID. It should accept your current password
4. Once in, you can delete the account, and enter your new Apple ID and password.
Note: if you are concerned about losing any information associated with the old Apple ID, then do a backup to your computer first.