There's only one thing that makes this dusty/musty smell, and that's mold growth. It is fairly typical for mold to grow in the evaporator drain pan and on the evaporator coil fins. The evaporator coil is located inside the air handling unit. If you follow the two copper Freon lines (one insulated, one bare) to the air handling unit(s), they terminate right on the evaporator coil. There should be a large plate that you can remove to access the coil and it's associated drain pan inside the unit. There's a drain on this drain pan that exits the air handling unit. It should be confirmed also that this drain is running free. The unit should also be pitched so that the drain pan can never have a low corner in which water will lay and grow mold during the off cycle. There are commercially available evaporator coil cleaners, which can also be used to clean the drain pan. These have a pleasant odor, and sanitize well. If the odor of Chlorox is not objectionable to you, you could use that as well. The cleaning of the evaporator coil is reccommended to be done on a yearly basis, ESPECIALLY if someone in the house has breathing problems like asthma. If this work seems beyond what you'd like to take on, contact a local HVAC company. They all have a flat fee for yearly maintenance on your equipment, which will include evaporator coil sanitizing.
OH, I just remembered. At HVAC supply houses they sell a tablet that you can put inside the unit right in the evaporator coil drain pan. This inhibits the growth of molds and such. Don't confuse the evaporator drain pan (inside the unit) with the auxillary drain pan (under the unit) which is normally used in attic units to contain any possible leaks from the unit. There's no maintenance required for an auxillary drain pan.