I've experienced this intermittently when I've purchased albums from iTunes that include a digital booklet with their purchase. The music will download to my device, but I have to download the booklet through iTunes on my computer, and until I would do so I would keep getting the sign in pop up dialog box on my device. Once I downloaded the album's booklet on my computer though, the sign in requests went away. Maybe try checking for any available downloads in iTunes on your computer and then download any content that may still be waiting to be downloaded, to see if that helps with your getting sign in requests.
Fyi, I'm not on any of the betas but I'm mentioning this as a possible scenario for what is going on with the sign in requests.