Thanks again to everyone who responded. Lots of good info and things to keep in mind going forward. The mechanic ran some “Frigi-fresh” through it. I was extremely skeptical about it but it seems to have, at the very least, cleared it up enough to where I’m not wheezing after 10 minutes in the car. I’m going to get the car detailed and replace the cabin filter and hopefully that will be the end of it.
Frigi-fresh is a BG product with a perfumed smell that is much better than the odor from a dirty A/C system. There are other treatments for dirty A/C systems however if the operator would selecting fresh air and not recirculate this would solve the problem in a few days.