Q: Changed Apple ID, can't sign in to FaceTime

I changed my Apple ID e-mail and have successfully updated everything (iCloud, iTunes, Messages, etc.) that requires my Apple ID on all my other Apple products. However, I CANNOT sign into FaceTime on my iPhone 6 running iOS 8.0.2.

Every single time I tap to add my new Apple ID to FaceTime (Settings, not the app), a pop-up keeps asking me to sign in with my old Apple ID.

1. When I sign in with that pop-up, it will tell me that the ID/password is wrong (obviously). (The pop-up won't let me edit the old Apple ID e-mail to the new one.)

2. When I tap 'Cancel' on the pop-up and try to continue signing in with my new Apple ID under FaceTime, the pop-up appears again. This prevents me from even being able to attempt to type a single character before the pop-up appears.

I'm following the same steps on my iPhone as I did with my iPad, but wasn't successful. I have tried turning off/on FaceTime and then signing in. I have also backed up my phone, reset my phone to factory settings, and then restored from backup with no luck. Setting my phone up as new is a last ditch effort I don't want to try unless I absolutely have to.

I have also followed everything in these two links that Apple recommends to do after changing your Apple ID:

Apple ID: What to do after you change your Apple ID

If you're asked for the password to your previous Apple ID when signing out of iCloud

Again, this is ONLY happening on my iPhone. I had no issues on my iPad Air (iOS 8.0.2) or Macbook Pro (OS X Yosemite). Please help!