Put on your rubber gloves and take everything out of the dishwasher -- not just your silverware and cups, but all of the racks. Use a clean rag to dry every surface, including the walls and racks.
Use your metal wool scrubber to wear away any rust on exposed metal or anywhere the coating has chipped. Then, go back through the dishwasher with that same rag, but blot the rag against the inner walls, in case some moisture has developed.
Use fine-grit sandpaper to wear down the edges on any rusted locations. Once all of the edging is gone, use a paintbrush to apply vinyl coating solution. You can find this at your home improvement retailer labeled as an "anti-rusting" chemical. Instructions may specify that the solution is for synthetic or metallic surfaces.
Cut any crumbling ends of rusty pieces with the wire snippers. Sand the ends and brush the coating on. Generally, if ends have come off, you'll need to add a thicker layer of coating in the repair, as snipping off the end removes the corrosion, but affects the overall stability of that piece.