The same thing happened to me when I first downloaded the app onto my Xbox Series S. While physically on the app only 3 letters would cycle through the shuffling. Like A through C or T through V, for example. After searching online for other users that suffered the same mishap, I came across your post. I have been patiently trying to figure out a troubleshoot for this (for you, myself and any others) and stumbled upon this ability to fix such issues. If you press the xbox home button and open up your Apple Music from the menu, where it shows up underneath Home and My games & apps, you can press start while having shuffled toggled and it will cycle through your entire library from A-Z, straight from the Home Menu.
If you have any questions or happen to misunderstand what I am saying, please feel free to reach out.
Enjoy!