The Alexa "Shuffle <Playlist>" algorithm doesn't work nearly as well as the shuffle in the Spotify App. It only shuffles through about the first 50 songs of my playlist instead of moving through the whole playlist.
I was able to solve this after finding some ideas on Reddit.
Create your routine, add music, then, next to "Play" put in the name of your playlist then "in shuffle mode" without quotes.
This will turn on shuffle mode as it plays. HOWEVER, it will not start on a random song! It always starts on the first song of the playlist, which was annoying until I discovered my new favorite playlist kickoff song "1 second of silence" - make this the first song in your playlist, and your routine will now start on a random song of your playlist.