Wood stain is not acid free and will ruin your fabric over time. Stained shelves/drawers are safe only if they are well sealed with urethane and since you say they will have to be cleaned with some special mix, I assume they have not been sealed. Whatever is in your mother's "special furniture cleaner" is most likely not acid free either.
You can easily line the drawers with Rubbermaid clear Con-Tact paper which is acid free, readily available and reasonably priced.
And as an aside, a well-known quilt preservationist that I have spoken with has no reservations about cotton being stored on or in plastic as long as there is ample circulation of air.