Okay, for some reason, I was running the Steam version through Proton (I'm on Linux) which is why the save file didn't automatically transfer. But otherwise I think the Steam and Itch version save to the same folder.
Hunting down the save files from Proton is always a bit of a hassle (for me they were in ~/.steam/steam/steamapps/compatdata/2570560/pfx/drive_c/users/steamuser/AppData/Roaming/SternlyWordedAdventures) but copying them over to ~/.local/share/SternlyWordedAdventures worked just fine - and should theoretically work backwards as well, if I want to play the full version on Steam when it comes out.