"Sign In to iCloud" password prompt won't go away, constantly pops up and won't accept password

I've got an iPhone 6 and as of noon yesterday I can't ge the "Sign In to iCloud" popup to stop popping up??

-I've created new passwords that work on the computer to login (hence I'm able to post on this forum) that when entered into the prompt do not work

-I've tried rebooting the phone and the popup keeps coming back

-I've tried to turn off/log out of iCloud but it won't accept the password I've created

-I've loaded the newest IOS update hoping that the popup was a glitch and that would correct it and it still didn't

-I've sync'd the phone and that didn't correct it

-I've tried to reset the phone through iTunes to the last configuration and it won't allow me to do that because I have to turn off the find my iphone feature - which requires a password that works on the computer but not on the phone???

Basically it pops up every 5-10 seconds, prompts me to enter my password, then won't take the password, then after a certain number of times popping up attempts to get me to change my password (which I do) then won't take my password???

Any ideas? I'm about to contact apple... the rest of the phone works fine and as normal but I'm continually disrupted by the damned popup.

Randomly when I mentioned it to my mom (we're on totally separate accounts and not at the same address) and she said her phone started doing the same thing 2 weeks ago and she's done all the updates and same trouble shooting as me to no avail?

Any ideas?