Anecdotally, I’ve noticed that this distinction is starting to get lost or muddled. I can’t give you specific examples, although there is a furniture shop close to where I live with the rather wonderful advert for “chester draws” (I do try not to mock odd use of English, but I rather like the inventiveness of this one, hence sharing it here).
To draw (a verb) is to make marks on paper with a pencil, or on another material with another medium, in order to produce a picture. It also means to drag something along or across (a horse draws a cart, we draw the curtains when we close (or open!) them); to reach a certain point (“we hope that the meeting will draw to an end by 7pm”); to work out (you draw conclusions); to attract attention (“his miming act drew a smaller crowd than he’d expected”) and various other technical things to do with pipes and sails and water.
There is a noun, draw, but this means a process by which a winner of something is selected randomly. (“We will have a draw of the raffle tickets at the end of the fete”), a game that ends in the same score for both sides or, in cricket, where the match has to be abandoned because it can’t be completed in the time allowed (“The match was a draw. Both teams got 1 goal”), or an attraction (“her burlesque act was a big draw and the variety show made a huge profit”). What it isn’t is anything to do with furniture.
A drawer (a noun) is the slide-out compartment in a piece of furniture, kitchen unit, desk, etc. It’s also a rather old-fashioned word for underpants, a drawer is someone who draws something, and you can be the drawer of a cheque when you write one. But the main use is the one to do with furniture.
“The winner of the prize draw received a beautiful chest of drawers. I’m going to draw a picture of it for the person who donated it, so they can remember it.”