So, I've only had steam for about 5 days, have don't starve together, it worked, then stopped (same issue as here). Take what I did next with a grain of salt.
Wanted the older version of the game (before auto-update) - so I went here: https://steamdb.info/depot/322332/manifests/
I then followed this guide: https://steamcommunity.com/sharedfiles/filedetails/?id=889624474
basically - I ran (from the terminal) - steam -console
It starts up steam - but with a console tab (become admin to access it)
Then from that console command line I did: download_depot 322330 322332 4273667276580057553
This is the version of the game from October 31st
It takes a long time, but eventually downloads to: ~/.steam/ubuntu12_32/steamapps/content/app_322330/depot_322332/
Then you go to the steam games directory: cd ~/.steam/steamapps/common/Don't Starve Together
And move all of those files to somewhere.
Then copy the downloaded files to this directory. cp ~/.steam/ubuntu12_32/steamapps/content/app_322330/depot_322332/* ~/.steam/steamapps/common/Don't Starve Together
Then, I go to steam, library, right click on the app, and mark to "don't update until I start the app", and I turn off networking so it won't be able to update when I start the app.
and it works again
So yeah, it broke with the new builds, but the October 31st build works.
I don't know steam or this app very well, so while I "think" what I did was safe, correct me if it seems wrong - hope this helps someone else.