You might have reached the 10-device authorization limit . Amazon only allows 10 devices to be authorized for downloads at once. You can deauthorize old devices in your Amazon account settings.
: Find the song, album, or playlist. Click the Download icon next to the title. how do you download music on amazon music
Tip: For entire albums or playlists, look for the (a downward-pointing arrow) on the detail page to save everything at once. You might have reached the 10-device authorization limit
High-quality music (HD/Ultra HD) takes up significantly more space. You can change your Download Audio Quality in the app's settings to "Standard" to save room. Download Purchased Music Using a Web Browser : Find the song, album, or playlist
Here is the complete guide on how to download music on Amazon Music for offline listening.
: Use the Find tab to search for a song, or go to Library to find tracks you’ve already saved.