Cardboard, paper, tissue paper, wood (all wood products) unless specially treated are acidic. This type of acid will slowly eat away at fabric. You can certainly use what you have, but you really need to think about lining the drawers. If the drawers are cardboard, probably the easiest would be to use contact paper. You could also line with fabric, but I think that would be harder. With wood you could apply a coat or two of polyurethane; not sure how that would work on cardboard, but might be worth a try.