Sounds like your first hvac company was ripping you off. Ask your new company to go to your coil and see if there is a leak. If there isn't the last company was just lining their pockets. If you have had ducts unhooked for a long time I would recommend duct cleaning, trust me I'm not a fan of duct cleaning myself. I have done hvac for 20 yrs. in GA. When I just bought a home built in 1996, we had all the ducts replaced immediately.
Duct cleaning is a band aid, like washing you hands is. Under your description, why wash your hands more than once? If it gets cleaned once, it shouldn't get dirty again. Well that is, unless they touch something dirty. Keeping your hands completely clean is virtually impossible. Same as keeping dust out of your home. You clean your ducts and over a 3-5 year span they get dirty again. Also, duct cleaners are not licensed but should be certified by NADCA(National Air Duct Cleaning Association).
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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.
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.