I've had this exact same issue which brought me to this thread. I, too, also had the problem where it would lose its place and a day later would go back and play 5-10 songs it previously played on the list. I also have issues where I would hit "Shuffle" on a playlist and I would hear songs repeated (sometimes several times) before other songs were played. The behavior used to be that I'd hit shuffle and it would play each song once all the way through the playlist until it got to the end, which at that time it would stop.
Up until today, I struggled my way through this by just hitting "Next" and hope it wasn't a song I already heard. The last straw for me was when I hit "Next" the exact same song was played. I couldn't believe it so I hit "Back" and it was the same song.
In my opinion, this is a MAJOR fail on Apple's part for this to have gone on as long as it has. I firmly believe that Apple Music is to blame. Well, I've abandoned the native iOS music player app and have purchased Ecoute for $1 from the App Store. Let me tell you, it's the BEST dollar that I've ever spent.
And Apple... shame on you. Steve Jobs would be super-****** about this gaffe.