What do I do about a VERY MOLDY dishwasher air gap?
You can simply take it apart and clean it with bleach water.  Most are usually designed for regular maintenance, bits of garbage can get pumped through there and may build up.  

If this is a rental, I would ask my maintenance personnel to handle the issue, you might put yourself at liability if you reassemble it wrong and cause water damage.  

Furthermore, I cannot say it would be causing you health issues.  But if you have a healthy immune system, it is highly unlikely the small amount of mold in the kitchen is dangerous.  Unless you are breathing regularly over the air-gap.