For Amazon Unlimited subscription customers, after you have added content to your watch, any changes you make to your downloaded playlist(s) will sync over automatically when your watch is charging and in the range of a saved Wi-Fi network1. You can also manually initiate a sync to update playlists and report all songs played to Amazon Music by accessing the "Sync" option from within the Amazon Music widget on your watch.
For detailed steps, see Related content below.
1The vivoactive 3 Music (Verizon) uses 4G LTE instead of Wi-Fi to sync playlists.