Musty odor ...

It may be from the overflow passage. This opening in the sink leads down to the drain (stopper). The passage often fills up with mucky soap scum which is a breeding ground for mold.

There's no good way to clean it out, but you might try a turkey baster filled with bleach solution to force it in the passage and kill the mold for a while.

It's also possible that mold is growing in the scum present in the drain just below the stopper and before the trap. I've seen some cases which resisted attempts to kill the mold until the drain was actually taken apart and swabbed clear of the scum.

Jim

You don't say where you live. If the sewer roof vent gets plugged it can force sewer gases back through the drains. This can happen for a variety of reasons, but if you are in a cold area it is especially likely during freezing weather with lots of precip.