Enabled 2 step verification then disabled it - now cannot sign in to iOS 6 iCloud account
I have a primary iPhone 5 64GB running iOS 6.1.4, and a work iPhone 5 16GB running iOS 9.3.1.
A few months ago I enabled 2 step Verification for my Apple ID and things were fine. I got tired of its erratic behavior asking me to verify too often and sometimes failing to send the codes to verify, so I disabled it. In order to do this, I was required to upgrade my old password to a stronger password.
Ever since then, my primary device (iPhone 5 / iOS 6.1.4) is unable to sign into iCloud. My new password is correctly entered, but it just keeps prompting for the password over and over. If I deliberately type the wrong password, it gives a different error, so that proves I'm entering it in correctly. The System Log shows invalid token for authentication. (While Photo Stream and iCloud Backup do not function anymore, oddly enough iMessage continues to work!)
Is there any way to fix this? I'm a very technical person, I have XCode, and am extremely detail oriented. I badly would like to get my iCloud working on iOS 6 again so I can back up everything before I consider finally upgrading to iOS 9 on my primary device. I can't backup to iTunes on my Mac Mini 10.9.5 Mavericks, because it fails saying in syslog MBDeviceBackup failed to copy keychain data. I tried disabling Encrypted Backups and it still fails the same way.
I considered signing out of iCloud on the device (so I could try signing back in), but this entails clicking "Delete Account" with scary warnings that make me afraid I'll lose something, even though it doesn't really seem like I will.
The Apple HT203055 article does not explain WHY this issue occurs. Is there some type of irreversible change that happens when you enable/disable 2 step Verification, that breaks iOS 6? Why wasn't I warned about this when I read the article originally informing me how to enable 2 Step verification?
Thanks for your help.
iPhone 5, iOS 6.1.4, 64GB