Your case just needs a diaper. I throw an old rag under the saw but all of my saws hold their oil quite well.
Folded over cut-off leg from old denim jeans works quite well.When pretty soaked (last one was in the case over 10 years) in the burn barrel it goes....They make great rags,not as soft or absorbant as old t shirts but for some uses are much more durable.