iPhone 5s Keeps asking for iCloud Password

A week ago, my iPhone began repeatedly asking for my iCloud password. (I was not doing anything with the phone when this started.) When I type it in, nothing happens. I get another request to "Sign into iCloud" and another and another. I've rebooted, updated to the latest iOS software, updated all Apps, reset network settings, backed up and restored, changed my iCloud password, and changed my Apple ID . I cannot access anything on the iCloud - calendar, contacts, iMessage, etc. If I do nothing my phone becomes warm (almost hot) to the touch and the battery drains. The only way to stop the madness is to go to Settings, click on my name, and sign out. The phone will then cool down and battery drain will stop, but I can't access iCloud. I have no problems signing into iCloud from iPad or Mac. How do I fix this? I feel like I've tried everything. I need to get into the iCloud on my iPhone 5s. HELP!