Thanks for this thorough response!
In terms of cleaning the ducts, we were told that they wouldn't withstand a cleaning (too old / thin) and replacing them would be very expensive.
We have a decent filter, but not something that thick. I'll look into this.
As far as how it runs, my understanding is that it only pulls from the return in the hallway, where the filter is. I'm not sure how stuff is sealed, etc., but at the same time I've done some electrical work in the attic and it's not so musty up there (hot, yes). I think the mustiness is in the ducts themselves. I myself don't notice it, so it's tough for me to pinpoint, but I trust my wife's nose.
As far as triggering event, my wife says she notices it when the air initially kicks on which, to me, supports it being in the ducts. She notices it when air has been sitting in the ducts and then enters the house, but not so much when it's flowing.
So it seems to me that our situation is likely duct-based, but that we can't clean our ducts. Given the mildness of this (my wife would agree) spending the money replacing the ducts is overkill. I was hoping the ionizer would provide a more moderate solution, but I just don't trust it, especially w/ a baby in the house.
Thanks again! And if you have any other suggestions, please let me know!