Not sure I agree that gaps are in the log runs but dust wouldn't be drawn into a duct it's a positive pressure (if it leaks it pushes air into the attack)
The intake side of the system has negative pressure which can suck dust into the system, the transition is where a filter goes.
That filter will stop the dust from being circulated.
You need to improve that not seal the ducts, I think they may be trying to seal your wallet$$$$$$$
First try a high quality air
see
If that don't work which I think it will then you move to an electronic air cleaner or a Ultra Violet filter, they are about $400 plus installation perhaps $200.