iPhone keeps asking for Apple ID password

I am unable to sign into my Apple ID, and it is preventing me from downloading any apps/music and making in-app purchases. If I try to make a download a song/app, free or paid, I am repeatedly prompted to enter in my password. I.e. I click "OK" and am immediately asked to enter the password again. I know the password is correct because if I intentionally enter the wrong password, then I get a message saying there was an error with either my account or password. The problem has been occurring since ~October 2014 after I updated my iPhone 5 to iOS 8.

Where the prompt occurs:

• When making a song or app purchase (free or paid)
• When trying to view my Apple ID in the App Store (bottom of the Featured menu)

  I am able to sign out and into my Apple ID, but again I cannot actually view my Apple ID

• When trying to view my Apple ID in Settings > iTunes & App Store > Apple ID

  Again, I am able to sign out and into my Apple ID, but again I cannot view my Apple ID

Solutions I have tried:

• Signing out of my Apple ID from within the App Store and Settings and restarting my phone (both hard and by powering off then back on)
• Erasing all content and data on my phone and restoring from a backup
• Enabling all cookies (Settings > Safari > Block Cookies > Always Allow)
• Automatically setting the Date and Time (Settings > General > Date and Time > Set Automatically ✅
• Changing my DNS settings (Settings > Wi-Fi > [chosen network] > DNS > 8.8.8.8)
• Confirming that the device is registered to my Apple ID by managing my account from a desktop
• Updating iPhone 5, MacBook, and iTunes software to the latest available and syncing my phoe

Tech specs:

• iOS 8.1.3

  Available storage: 7.6 GB

• iTunes 12.1.0.50
• OS X Yosemite 10.10.2
• Two-Step Verification is not enabled

Thank you for anyone who is able to offer assistance.