An update to my problem:
I did find a fix for my problem, though my issue was that no Apple Music was playing but all purchased music was.
To fix (****see my note below before trying!!):
I went to Settings -> Music and uncheck the box next to iCloud Music. After a few seconds, I re-enabled it and got a message to have my music merge or replace my current music (or something to that effevt - it was a few days ago). After doing that, I went back to the Music app and it had to resync all of my playlists again. Once they loaded, everything played nicely.
Again, I have no idea why this worked, but it solved my problem of Apple Music not playing at all. Hope this helps someone.
****Also, I should note I have iTunes Match as well, so not sure if that's why my playlists reloaded. It's possible doing the steps above, you may have to rebuild all your playlists. I didn't play around with it much to figure out how it happened, I just know what I did and it solved my problem.