I quite agree with both Eric and Stallone. In fact, to me, from a British perspective, both of these words sound very old-fashioned, but especially 'drawers'. As I remember, drawers generally referred to women's clothing and underpants to men's. Personally, the only time I use the word 'drawers' is when referring to the place where I would put my 'pants', i.e. in my chest of drawers! (Same word different meaning). I remember a very old joke (my grandparents generation...): people would sometimes say at the approach of winter "Ah! winter draws/ drawers on": with the double meaning of winter is arriving/ time to put on some warmer undergarments (different words same pronunciation). I just did a search for that expression and found that it was used as the title of a song from 1935 by Gracie Fields, so presumably that is where the joke originated. You can find it on Y/T, if you have nothing else to do! ;). - I think the accent is probably from Yorkshire, England, although I might be wrong.

As Stallone rightly said, "drawers" is not common today, in fact I would say it is only really used in a joking way, because it is an older way of referring to underwear. I say underwear because "underpants" is also not the typical word. In North America I would say "underwear," or the specific name of the specific item of underwear you're speaking about, is most common. In Britain, they use "pants" to mean underwear.

Similar meaning but "drawers" is less common today.