Food safe varnish or paint
In parts --

The smell from any paint or varnish is a result of outgassing.  Leave the doors open and that should disappear in a week or less.  If it doesn't, or the paint remains "tacky" then strip and redo with fresh paint.

Unless you are storing foodstuffs unbagged, there's no direct contact between food and the drawers, so it's a nonissue.  If you are storing loose items (onions? ) I still strongly doubt any significant material transfer can take place.  If worried, peel and/or wash before use.