iCloud sign in loop

Operating iOS 8.3 on my iPhone 5, I am getting a continual popup message asking me to sign in to iCloud. This happens about every three seconds. The popup asks me to enter my Apple ID password. I enter my password and it tells me it is incorrect. Using my phone, I reset the password using the email option and follow the link to set up a new password. I set a new password and then set my phone down for few minutes. Picking it back up, I unlock it and wait for the popup to appear, which it does, without fail, within seconds. I enter the new password when it prompts me to enter my Apple ID password. It again tells me that this password-- the one I have just reset-- is incorrect. I have tried to delete iCloud from my phone in order to stop this, but it, too, requires that I enter a password in order to delete "Find my iPhone."

Lame, apple. Please, someone offer a solution. Thank you.