I've been having this issue for a long time, and running as an administrator does not fix it for me, neither did reinstalling it, or changing the default audio device for Spotify in Windows settings. This is a huge issue and it's been happening for years. Seems the Spotify devs are unable or unwilling to actually fix the root cause. If it persists I'm going to cancel my Spotify premium account.
I have the very same problem on windows 10, song skips and skips and none of them play, not event the one I've previously downloaded (not even in offline mode). I tried uninstalling the app and installing it again from microsoft store but it didn't work.
I've just found this workaround to run the app as administrator and it worked, but if I run it again not as administrator the skip problem happen again. I don't want to run the app as an administrator and have a security risk!
I'm really furious about this problem and the constant updates that can lead to this kind of problem. The updates must be optional, not something I have no opinion specially for this kind of problems... and of course, if I ask for help to support they are gonna ignore the problem and say my computer is the problem.
I would like to not believe that this is device's problem... Because I have used Spotify for a long time as free user, and then I didn't have any problem ever. But just yesterday night I switched to Premium - and this morning IT started to happen with my desktop app, like you say.