The original furnace was a coal fired “octopus” design which had no blower to circulate the air but relied upon warm air rising to distribute the heat. That also means that the system breathed air in from the basement and most of this type did not have a cold air return system. At some point (probably in the 1940’s) the system was upgraded to a forced air gas furnace. The upgrade included replacing the 12″ diameter air ducts exposed in the basement with 6″ ducts and a cold air return system of ducts from the two (only two) air returns on the first floor.

I was with you until you said avoid steaming cleaning or moisture, there is no way you can remove mold or any other type of biological without moisture. The best way to do this is in fact with a steamer using a commercial disinfectant and a non reactive odor remover so your home smells refreshed and not like a hospital . This is directly contradictory to your article.

Manufacturers of products marketed to coat and encapsulate duct surfaces claim that these sealants prevent dust and dirt particles inside air ducts from being released into the air. As with biocides, a sealant is often applied by spraying it into the operating duct system. Laboratory tests indicate that materials introduced in this manner tend not to completely coat the duct surface. Application of sealants may also affect the acoustical (noise) and fire retarding characteristics of fiber glass lined or constructed ducts and may invalidate the manufacturer's warranty.

I recently replaced my 44 year old furnace. Over the years we have had several construction projects inside the home. For many years breathing the heated air from the duct work caused excessive sneezing, sinus issues, and lung irritation. . The heat smelled like dust. To maximize air quality I added a humidifier, MERV-11 filter, and UV light onto the new furnace. To further improve air quality, I had the ducts cleaned and sanitized. I took before and after pictures of the inside of each return and output duct. They were filthy before, and sparkling clean and fresh after the procedure. Now when running the system, clean sanitized air flows through my whole house. A huge difference. It will make sleeping and being indoors during the winter months far more tolerable. I use window air conditioners, and run the system in the summer to keep the house air clean and fresh on hot summer days too. It works! Great decision!
The short answer is yes, duct cleaning, done right by a reputable company will help with the smell. It is likely though the smell is coming from another source, ie the crawl space. It could also be that another rodent is dead in the duct work. A visual inspection would be in order to check for any breach in the duct system. A duct pressure test would be able to tell you how much air is leaking from the system over all. A hole large enough for an animal to enter would loses a great deal of air, over 300 CFM. It would be worth looking into having this test done and it should cost around $200. Cleaning the air handler and evaporator coil will greatly improve the performance of the air conditioner. Dirt and dust on the evaporator acts as insulation and reduces the amount of cooling done be the coil. Any quality duct cleaning should include cleaning the air handler and coil, and the tech should allow you to watch him perform the work as it is done.
You should be aware that although a substance may look like mold, a positive determination of whether it is mold or not can be made only by an expert and may require laboratory analysis for final confirmation. For about $50, some microbiology laboratories can tell you whether a sample sent to them on a clear strip of sticky household tape is mold or simply a substance that resembles it.