This app was mentioned in 11 comments, with an average of 2.82 upvotes
No problem. I'm not sure which app you're referring to. Are you using an Asus Flip on Dev channel with Android apps and some sort of Amazon Music app? No other Chromebooks have access to Android apps at the moment. Wasn't sure there was an Amazon Music app in the Google Play Store.
Oh, yeah - now I see it. Here.
So the above linked app on a Flip on Dev channel, right? If so, I'm not sure how to fix it.
Otherwise you're out of luck until Android apps are rolled out to other Chromebooks.
I've been getting my permanent non-DRM mp3 files from Amazon MP3, but I do subscribe to Google Play All Access ($10/mo) and you can basically play any song you want at any time, and synchronize to your device (even sync "radio stations") to listen offline or away from WiFi if you're one of those people who are out of data after the first week.
To get the MP3 files from Amazon Music (for music that you've purchased, you can't download MP3 files of tracks that you're just streaming), download the Android app on your Chromebook, then download the albums/tracks that you want. The MP3 files will be downloaded to My Files > Play Files > Music in the Files app.
As for getting the MP3 files to your MP3 player, that depends on what kind of player you have. Ideally you'd be able to plug it in and drag the files over, but it doesn't sound like that's working for you. What model of player do you have?
FYI: New version of Amazon Music was released today. I have experienced this issue, but have not tried the updated app yet. Hope it fixes it.
https://play.google.com/store/apps/details?id=com.amazon.mp3
Not the Web UI but it's a start; The update now has Alexa.You can now cast to your Alexa devices. Use your phone to play, skip, shuffle, and more on Alexa and Chromecast devices.
https://play.google.com/store/apps/details?id=com.amazon.mp3&hl=en_US
You also get Prime Music! It sucks but whatever it's free. I still pay for Spotify because Spotify's library is about 10 times larger (or more).
https://play.google.com/store/apps/details?id=com.amazon.mp3&hl=en
Upload and stream from the cloud services:
Amazon Music - 250,000 songs for $25/year
Google Play Music - 50,000 songs free
Apple Music - 100,000 songs for $25/year
Stream from your PC:
Stream What You Hear - http://www.streamwhatyouhear.com/getting-started/
Muzecast - http://www.muzecast.com/
Airstream - http://airstream.io/connect-pc/
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They do have Amazon Prime Music on the Google Play Store.