The sawdust, if I understand your question, is caused by not cleaning the tracks and not sealing the wooden drawer boxes. You can simply paint the cabinet's unfinished interior and the drawer boxes to keep the wood from flaking off. Whatever you do, avoid painting any of the hinges or drawer tracks, or you will wish you never asked for my advice, assuming you are still reading this after the crack about the floors, no pun intended. You may want to seal the wood with polyurethane, but make sure the pilot light is off if you have a gas stove and ventilate the room by opening windows, even in an adjacent room.
If you do not have metal drawer tracks and the wood drawers slide in wooden tracks, linseed oil can be used to seal the wood. This penetrates the wood and helps to reduce the friction that may be causing the sawdust. Wax polish may also work in the wooden tracks. Just make sure you have floors in your building before you get started, so the neighbors down below don't complain of funny tasting soup when you start to paint. Good luck!