I'm having the same problem as well. The issue occurs only when the web version of Spotify is used (or when you control the web version using the Spotify smartphone app). Trying Incognito gives nothing, so the problem persists.
I turned off my adblocker, no luck. I tried going incognito, that didn't work. I tried clearing my cache then I cleared my cookies, those didn't work. I couldn't download the app because it's incompatible with my chromebook. The app works fine on my phone, and the web player used to work fine on my chromebook, as I was using it around week ago. Now it skips songs after 2 seconds, even if I pause the song it still skips it, but it plays ads all the way through. Hope this info helps.
EDIT: I'm using a Samsung xe303c12-a01us Chromebook with some version of ChromiumOS (ChromeOS) derived from Linux? and I have developer mode enabled and Ubuntu installed.
This started yesterday for me too. I tried using an incognito browser and using the downloadable Spotify web player but it keeps doing it. My Chromebook doesn't support Android apps so I can't try using the app.