Q: iMessage automatically wants my old Apple ID...

I recently upgraded to an iPhone 5s and changed my Apple ID email address to my primary email. However, I kept my old email address as the secondary address to my account (just in case). I've logged out of my Apple ID account in iMessages, iCloud, iTunes / App Store, and Game Center on my iPhone and tried to log back in with my new email. It works fine in iTunes / App Store and iCloud but iMessages and FaceTime seem to have a problem.

When I go to Settings > Messages > Send & Receive, it says that, so far, only my phone number can receive iMessages and asks to add an email or Apple ID. When I try to do either option, it automatically prompts the password for my old Apple ID. I tried typing in the password for the old and new account. It tells me the old password is incorrect. When I try the new password, it just consistently refreshes and asks for the password again.

Does anyone know what's going on and how to fix it?