If you go north of the main store in the game you will find a biogas plant. This is slightly north west of the pig farm. If you go here, you will see a large red bucket near the main gate. This is where you sell slurry.
You will first need a slurry tanker (not a spreader). I think you need to use a semi to pull these as they are large. Fill it up with slurry and drive it to the plant. Make sure the rear of the tanker is next to the red bucket and you will see the option to unload the slurry.