Since you have forced air heating, buy the best (highest rated) filter element for the furnace. Make 100% certain that the filter fits perfectly (no air can leak around it).
And....have a UV-C biocidal lamp installed in the return duct (or plenum). They are made just for this purpose and kill mold/fungal spores and also bacteria/viruses.
You will have less invested than in one very good room air cleaner. If you find that a particular room (bedroom maybe) needs even more attention, then get a stand-alone model for that room. They, too, can be furnished with UV lamps.
If you suspect that chemicals (rather than spores) are the causative factor, room air cleaners can be had with high-volume activated charcoal as well as HEPA filtering.
Jim