I doubt Febreze would hurt it and it might work wonders. I think what I'd do is use the Febreze and let it dry, then sprinkle baking soda in the bottom of the drawer and close it up for a couple days.
I found a few links that might give you some ideas.
http://www.thriftyfun.com/tf201130.tip.html
http://davesgarden.com/community/forums/t/647584/