I was having the exact same issue. Maybe this will help...
Within the app, in the playlist, you should see the play button, the share icon, and the three dots.
If you don't see the share icon, the list isn't in the Amazon cloud (not only do ALL the songs have to be in the cloud, the list itself needs to be uploaded, once all the songs are in the cloud, upload of the list takes 2 seconds at most).
Clicking the three dots, if you select upload, do you get an error message?
If so, look through all your songs, make sure every one of them is "Available Offline and in the Cloud" - it shows as a little computer screen icon with a check-mark on it.
If one song is not uploaded and giving you issues, try to slightly change the metadata (right click on song), and retry the upload to cloud option for the song.
Good luck!