LOL! that was my one odd chore I just completed. I found a pack of 4 velcro adhesives and 16 felt pads and proceeded to complete my set of 8 chairs. Some had been off for over 5 years (prior home had carpet) and we just never used all the chairs a lot, so I put up with it, but then noticed last week the floor scratches... I guess they do work, indeed.
The ones with the push-in nail post are the best. Our have lasted 5 years with light use, and I will try to squeeze out another year. I guess I could look to see if I could just replace the pad.
The best bet may to to buy a large package for your drawer, a package of double sided tape, and just replace as they fall off.
ETA -- Cardboard is one of the scratchiest things on wood floors! Snacky - It only protects (cushions against depression marks) if you don't move the chest.