DIAGNOSIS That musty smell is mildew growing on the cold, wet evaporator coil deep inside the unit—a common issue when it's shut off damp or used in high humidity. Rinsing the filter won't reach it.
FIX You'll need a coil cleaner. Unplug the unit. Remove the air filter. Look inside at the aluminum fins of the coil. In a well-ventilated area, spray a generous amount of non-rinse, evaporator coil cleaner (found at HVAC or appliance stores) directly onto the coil fins. Let the foam sit for 10-15 minutes as directed—it will dissolve the grime and mildew. Then, set the unit to Fan Only mode for 2-3 hours with the filter still out to completely dry the coil. Replace the filter after.
IF THAT DIDN'T WORK If the smell persists, water might be stagnating in the bottom drain pan. With the unit unplugged, remove drain plug and tilt the unit over a tub to drain any standing water. You can flush the pan with a little diluted vinegar via the drain hole, then drain it again to disinfect.
HEADS UP Never use harsh household cleaners, bleach, or pressure washers on the coil. They can damage the delicate fins and the special hydrophilic coating. Only use products specifically labeled safe for AC evaporator coils.