iCloud constant password request/can't sign out/Cant sign in mail

I've had my iPhone 6 for nearly 2 years and this problem has occurred recently, it is updated to iOS 10.2.1. My phone is constantly asking for the password of my Apple ID which it accepts, but asks for again a few seconds after. I've restored my phone and put my last back up on again but the problem is still there, I've also changed the password and logged out of iMessage, FaceTime and the App store which all are fine logging in again. I also can't log out of the iCloud as I press 'sign out' then 'delete account' but this results in a message box saying 'Couldn't delete account This account may be syncing data please try again' which it isn't as it is not plugged into iTunes, but if I try again, a grey 'signing out...' box appears and the screen goes back to the settings menu, however the log in loop starts again.

I'm not sure if this is linked to this issue but my mail app has also had problems during the time the iCloud password problem, all the mail accounts that were signed in have disappeared and when I go in the mail app, there is a welcome screen with different email servers, if I try and sign in the message box says there is an identical account signed in but in settings there are no accounts signed in.