My first visit to Pedants' corner but this has been irritating me for some time. It seems that whenever someone offers this item on a local sales site (lots of them on Facebook) or for rehoming on Freecycle the drawers are usually referred to as 'draws'. I can't help it - it causes me to shout at the laptop, which does no good, of course!
Passports not in the drawer I always keep them in. Turning the place upside down.