My disappointing "Auto-play on Music App" experience: I tried literally everything to turn off auto-play, spent (no kidding) hours researching and trying different "work arounds." I sent pleading e-mails asking Apple to give us a "Turn Off Auto Play" option via https://www.apple.com/feedback/ (which is the official place to send requests to Apple.)
No solution.
In a "scorched earth" attempt to fix the problem, I finally just cancelled my Apple Music subscription, (and sent an explanatory e-mail to Apple Feedback, stating that as a result, I would no longer pay $$$ for an Apple Music subscription until they fixed the problem.
The result? No more Auto Play! (Yaaaaaayyyy! Snoopy Happy Dance!)
Downside: It was a Pyrrhic victory since I now have to load music onto my phone. However, having music & podcasts automatically blast into whichever bluetooth device is connected was driving me nuts, and was really embarrassing during business calls/events.