Musty chest of drawers

I have a chest of drawers which must be about a hundred years old. It looks attractive, darkish wood in fairly good condition. However, it was stored for years in a damp, mouldy building and it smells a bit musty. I want to use the chest to store clothes but would like to give the drawers a good clean and hopefully get them smelling fresh. It's now completely dry and I'm keen to start using it. I have googled and looked at various treatments and am thinking of trying white vinegar, or maybe tea tree oil, but not sure I want my drawers (wooden) or my drawers (cotton) to smell of vinegar... Has anybody dealt with this problem? All suggestions much appreciated.