Sounds like there's a mildew problem. Easy way to find out, takes maybe five minutes to set up, take the drawer out on a nice sunny day, put a bottle cap full of Clorox in it and cover it with a plastic trash bag, making sure that any painted or finished surfaces are outside the bag. Just let it sit in the sun for a day, remove bag and bottle cap. Sniff, if you like the feeling of your nose being burned, air out and if it was mildew the smell should be gone.
If it's mildew, you have to kill the mildew and the clorox will do that nicely, also anything else that happens to be living in it. Chlorine is not good for living things, kills the mildew and removes the smell at the same time.