So, if you have any *real* suggestions on how to get the musty smell out of
the dresser, I would appreciate it.


Sounds like mildew. Spray white vinager inside and around the
outside of the drawers. With the drawers out, spray vinager inside
the case. Leave it in the garage for a day. Not too heavy a spray.
Alternatively, wipe it down with a damp vinager sponge.

Lysol might also do the job and there has to be commercially available
mousemilk products at the hardware stores.