Use a scrub brush, dishwashing soap and warm water in the laundry/bath tub first. If that doesn't work try an SOS pad with a dry bleach type product like Ajax or Comet. Last thing you want to do is replace the cork - BIG bucks
I'd second the spendy part. I'm the only commercial game in town up here, and ~attempt~ not to get stupid with pricing. Except for corkwork. The work is not hard, or technically challenging, but spendy for what it is. I request 4.00 an inch for repair corkwork, a little less for fresh, as it's in the package price of a new rod.
And folks are happy to pay that, as it's lower than other places, and no mailaway costs or delay in the transport system.