Mold is a tricky thing, once it gets in the wood it is very hard to get rid of it. I wonder if you can soak the flutes in bleach, I saw very good results with bleach. The only downside is the smell and the staining.

A dehumidifier might help in those moist rooms. It can help by removing the excessive water out of the air.

It's awesome that you were able to rescue your flutes. Mold....the very word gives me chills. It is very dangerous for the lungs as well