Like many, I have been experiencing Apple Music songs skipping in my playlists, one of the many delights of the new app. Many songs were greyed out in the playlists and did not play. 

I don’t know if this will apply to anyone else, but I found a solution that works for my system.

I remembered that for some reason I had four separate Apple Music systems showing up as “connected accounts” in Service Settings -- even though they all connect to the same Apple Music account, same login and password. (No idea how that happened, but that had been the case for many years. It had never caused an issue until recently.)

I tried to delete all Apple Music systems except my Primary Account. The app did not allow that (why? who knows!).

So I reauthorized each of them, keeping only one as my primary account.

That has solved the issue with my playlists - all my Apple songs now play!

Note, I previously tried rebooting all my speakers, router, deleting and reinstalling the app, etc, with no joy. 

TLDR: If you are having issues with skipping songs, check your Connected Accounts in Service Settings and reauthorize all the systems that show up there.