How to clean a couch?

I recently bought this secondhand couch and thought it would be a simple throw-it-in-the-washer type cleaning, but I understand the cleaning code is ‘S’ and that isn’t the best course of action for it. As you can tell it has some serious discoloration on the seats and some pilling. I NEED it to be fully clean before I can enjoy it, so what do you guys recommend? I feel like the “dry cleaning solvent” that is suggested won’t get it as clean as I want it. Do you think using a wet vac on it would work and not destroy it?

Also, please ignore the clutter in the background, I just moved.