Again, I am posting; adding to my previous post.
I haven't had any success signing out of iCloud or the Apple Music service when the problem is apparent. When I do attempt to sign out of either of the applications, it hangs and ultimately stops trying after about 5-10 minutes.
The good news is that I haven't had the problem happen in over a week. Each time I access apple music, I can play anything I want. Something I noticed; when I am having the problem, if I go to Apple Music / Account info (the little head icon while in apple music). Manage Family... is not in the list.
After I correct the problem or when the software is working properly, when I navigate to this location, I see 'Manage Family...' as I am on a family license.
I didn't do anything to resolve my issue, possibly something changed on Apple/Server side or something, but I am glad.