Mold/mildew odors in forced air heating and cooling systems are annoying and can be a real pain to locate and cure. Rather than standing water causing the mold, it’s likely a condensation point either on the coils themselves or somewhere along the ductwork.
Gas heating systems with air conditioning are not immune to this sort of thing however, unlike heat pumps, tend to receive more regular cleanings as a part of their routine maintenance than heat pumps seem to get. Many folks apply the “if it ain’t broke, don’t fix it” principle to heat pumps, mistakenly ignoring such virtues as cleanliness of filters, coils and ducts. Be sure to clean the whole air-handler itself, including the coils.
Wads of dust can become lodged between the fins of the coils and get wet during air-conditioning operation from the system condensing moisture out of the air. Turn the system off and let it set a while and mildew sets in. Turn it back on and the mildew dust-bunnies act like an obnoxious air-wick releasing the mildew smell.
Unless you are one of those very advanced and dauntless do-it-yourselfers I don’t recommend tackling the job yourself. Call your service provider and have the system cleaned and the coils vacuumed or air-blasted. Clean and make sure the whole air handler chassis gets a good going over. Make sure the heating coils get looked at, too.
Check the condensate pan located under the air conditioning coils that catches the dripping condensate to be sure it’s clean and does not let water stand in some part. I have seen primordial swamps form in these pans.
Cleaning the whole duct distribution system is a good idea if the ducts are seriously dirty. Take a flashlight and a hand mirror and examine as many of both the return and supply ducts as you can get to. Pull the registers up from the floor and unscrew the grilles.
If you have a good vacuum cleaner with a wand and brush attachment, you can do a pretty passable job of duct cleaning yourself. Hiring a professional duct cleaning service will give you a whistle-clean job but be prepared to spend a couple of hundred dollars for it.