How to get that musty smell from the old wood cabinets?

You can put some fresh ground coffee into small cups and sit it inside the cabinets. Close the doors and let sit for a few days. That should help absorb a lot of the musty smell.

You can also put newspaper in it. :)

I have read that containers of charcoal will absorb odors. Can always add a new coat of paint.

Try placing newspaper on the bottom shelves. Add a lot of charcoal used for your grilling. Close doors and try not to open too often. I purchased a very, very old, smelly pie cabinet that smelled moldy and used this technique, keeping doors closed for about two weeks. I then painted the inside with whit shellac then a nice cream color. Have it in my townhome and absolutely no smell. Hope this helps.

Charcoal in them for a few days, put some in each drawer. Lavender works too. And wipe the inside of the drawers with a cloth dampened with vinegar.

Wadded up newspapers to fill the cabinets. Saucers of vinegar. Unused charcoal briquettes (not self-light). Leave any or all of these sit in the closet without disturbing them for 5-7 days.

Have you tried cat litter placed inside and with doors closed? It is easy to get out and can be used for other odors as well. I also carry in the trunk of my car during the winter in case I get stuck on ice.