Have you tried:
Settings>General>Reset>Reset Network Settings
If that doesn't work, try
Settings>General>Reset>Reset All Settings
If that doesn't work, then a Backup and Restore may be needed. If you go the Back & Restore route, after you have backed up your device, when you do the first Erase All Settings & Content, do not set it up from the backup - set it up as a new device, then try signing in. If that works, then Erase All Content & Settings again, and set it up from the backup. If it still works at that point, you are good to go. If it doesn't work, then that would indicate that there is some corruption on your device, and you will have to "rebuild" it manually (not from the backup).
Best of luck,
GB