Update: I've found that deleting and re-installing the Music app has resolved this problem for me, at least for a couple days. I am hesitant to say that the problem is fully resolved, both because (1) others have indicated that it sometimes appears again after an additional iOS update and (2) a soft reset (the reboot from simultaneously pressing the sleep/wake and home buttons) had previously appeared to fix the problem, but then the problems with Music playback appeared again after a few hours.
Also, some notes that Apple doesn't highlight regarding their suggested "fix" of deleting and re-installing the Music app:
First, be sure to go to the Settings menu and write down your Music settings before doing this, because your prior settings won't carry over after you've re-installed the app.
Second, it appears that quite a bit - but not all - of the downloaded music on my phone was there after I re-installed the app. Downloaded music that I'd previously ripped from CD's or purchased from iTunes appears to be there in full. I also, however, have an Apple Music membership and manually download to my phone some but not all Apple Music songs that are in my Music Library (due to storage space limitations). From what I can tell, none of these Apple Music songs downloaded to my phone again even after I turned on iCloud music library, meaning that these songs/albums appear as "available" songs in my Music Library but aren't downloaded to my phone. This may not be an issue for anyone who turns on Automatic Downloads and therefore downloads everything in their Music Library (including Apple Music songs) to be available offline, but it does appear to be a problem for anyone who wants to have offline access to some but not all of their Music Library songs that are from Apple Music.
Again, this highlights for me that Apple needs to apologize and promise to fix the underlying issue rather than presenting sub-optimal work-arounds as the "answer" to this problem that Apple clearly caused in recent iOS updates. These work-arounds not only take up the time of users but also, based on my experience, have the potential for unforeseen consequences, particularly for people who have decided to set up their Music Library in a very specific manner.
(As a bit of background, my phone is an iPhone 5s. I update iOS essentially right when updates are released, and the music problem of garbled sound, cutting out, etc. did not occur on my phone until sometime in July. My iPad Air 2 has not had the same problem.)