Not that I've dealt with accordian cases per se, but here goes:

Take/pry the "icky crushed velvet" lining out of the box, lay it out on a large piece of paper and cut yourself a pattern from it. Then get some new fabric (velvet is very pricey, perhaps velour is more affordable) and use your pattern to cut a new liner. (I wouldn't expose the new one to the old directly, or it might make it stinky also) Meanwhile, it all of the time it will take from when you eviscerate the box until you finally finish relining it (takes me weeks or months to sometimes finish sewing projects, hence my thoughts of the passage of time) you might think of something like a liberal sprinkling of baking soda to soak up the rest of the odor from the case.