Wipe them out with vinegar. Place baking soda, newspaper or coffee grounds in the drawers.
I know there are lots of suggestions out there but I've struggled with this myself and nothing worked. I'm at the point I considered painting the dresser inside and out to seal it.
I had this problem with a vintage dresser. It did not have a musty old smell, but the drawers reeked of perfume. I tried soap & water, then vinegar, then baking soda, coffee grounds, sitting in the sun. None of them worked. Finally, I coated the drawers with a layer of shellac. It goes on easily, dries quickly, leaves no odor of its own and has not lost its sealing power for 8 years, so far.