Musty smell, only in drawers in the basement dresser

How dould I get the musty smell out of two dresser drawers in the basement laundry room, and how do I put textile bags back in the second drawer without them starting to smell again? And could I do this in the winter?
I have a townhouse with a basement, with a family that is quite comfortable (not humid, not too dry, not too warm, not too cold), and a laundry room I use for laundry, and for storage of paint, etc., camping equipment, scrap wood, scrap metal, etc, and I have a work bench with a couple power tools.
Under the work bench is an old wood dresser. The top drawer is for tools, which do very well there, don't rust or anything.
The bottom two drawers I have just stored carrying bags of various sorts, usually that I got for free, made of canvas, vinyl, leather, etc. Nothing expensive, unless it is quite used. But I like having a bag for every purpose.
These two drawers have been musty for more than a decade. Is one particular material in the drawers likely to be responsible for the musty smell. It infects everything, and when I actually wanted to use one of the bags, it smelled too bad to use in public.
Maybe I shouldn't have, but I've kept the two drawers closed almost all the time, and nothing else in the basement smells like this. Nothing else smells bad at all. In fact, on a couple occasions I leave the laundry in the dryer, still slightly wet, and it doesn't even mildew. The first time it was there for 2 days, and the second time for 5 days. The dryer is on the opposite corner of the laundry room from the dresser, but it's still not very far.
I've run out of space in the house, and have to use one or both drawers for tool storage. (I avoid buying big tools, but I've needed a lot of small tools for various projects.)
I would like to sort out the bags that can be made to smell good from the ones that are doomed. Most can't be washed, although some could be put in the dryer. What I would probably do is wait till summer and put them out in the hot sun. The heat and UV do good thigns.
But how would I get the smell out of the drawer, and how could I put these things back in the other drawer without them starting to smell again? And could I do this in the winter?
Thanks.
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wrote:

<snip>
Try putting a bag of charcoal or box of baking soda in the drawer. Think of a way to do this without dumping the charcoal dust or soda powder. Putting the drawer out in the sunlight is a good idea.
Another fix is to clean out the drawer, sand it, then give it a spit coat of shellac. Allow a week before putting the drawer back in service.
Not my personal solution, but you can try to mask the odor by putting a lavender sachet, a few bars of soap, fabric softener sheet, or a bag of pet cedar chips in the drawer.
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I should be able to do this without dumping. :)

Wow, these all sound like good answers.
Lavender might be a bit much for a manly man in his work shop, but soap sounds good. I've heard a lot of good things about fabric softener sheets (I used to have a free sample, about 10 or 15 years ago. I wonder what happened to it.) and last summer I had a bag of cedar mulch. It smelled great. I used it for mulch, but I can buy another bag. (I also made some cedar pickets for my neighbor's fence, and it's been a year and they don't smell anymore. Although I broke open one of his original pickets (25 years old) and it smelled great too, as if it were new I think.
But though these things smell good, baking soda and charcoal are famous for absorbing odors, "not covering them up" as the tv commercial says. Charcoal is used as a odor filter commercially. How come I couldn't think of any of these things?
My gosh, I bought some Campbell's soup a couple weeks ago, and they are trying to catch the quick-cook market, so they now have some of their soups in microwaveable bowls. No need for dilution with milk or water. (already diluted, or never condensed) So you get about half the soup for twice the price. But it came in container with a lid with some holes. I save old containers and didn't have one of these so I saved it, and it will be great for putting in the baking soda or the charcoal.
Thanks a lot.
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