Before considering something as much effort as refinishing, I'd scrub all surfaces inside and out with Murphy's Oil Soap, rinse, hand dry, and leave in the sun until thoroughly dry. And I do mean ALL surfaces top & bottom, inside and out, not just the easily accessible outside bits. Do this a couple of times and I'd be surprised if you didn't find most if not all the mustiness has gone away.
You might also want to check to see if there isn't a bit of mold on the bottom of the feet where the furniture was in contact with a damp floor. Flipping it over to check this while cleaning all the bottom bits will give you an idea of any places mold might be lurking.