Adding a new account to my Samsung phone
There might be a third party app that's in your way. Therefore, try adding the account while in safe mode. Because while you're in safe mode, all third party apps are disable, so they can't make any trouble.

To go into safe mode, turn your phone off, and then turn it back on while holding down on the volume down button. When the phone turns on, it will say safe mode on the bottom left of your screen.

Now try adding the account while in safe mode. Go to settings &rarr; accounts &rarr; and account &rarr; Google....