iCloud sign-in won't stop popping up.

For the past month or so, an iCloud message requesting me to sign up has been constantly popping up. I updated my phone to see if it would fix it: it didn't. I know that it's not anything wrong with my iCloud in general, because it works on my other devices, it's only doing it on my iPhone 6s Plus. I've changed my password that I've had for years hoping it would solve the issue: it didn't, and now I hate this situation even more. I've tried signing out of iCloud but all it does is have the sign out greyed out after I put in my password to turn off FindMyIPhone. I've also reset it way too many times (at one point I almost lost everything because it got stuck on the Apple logo, but I fixed that by connecting it to my computer to update it, totally different topic altogether but that hasn't happened since; this is near the time it first started happening). I can't just reset my iPhone because I was planning on backing up until this happened and the last back-up for this iPhone was last year (I know, that's bad, don't judge). This is super annoying and all the articles that I have run across are like a year to several years old so they don't really help me now, even those articles' comments said that their ideas never worked out in the first place either.