> Maybe if this keeps up, I'll just alternate between having and
> not having the filter on days when my allergies aren't bad.
There is no good reason not to a) use both filters b) all the time.
No good reason not to.
Depending on your indoor environment (and if you have allergies):
Change the disposable (white) filter once or twice a month.
Rinse out the foam (gray) filter weekly.
Wash it with a gentle soap once a month.
Must be completely dry before using again.
And, yes, for the disposable (white) filter the "fuzzy" (soft) side is to the outside and the "plastic" (firm) side is to the inside of your machine.
Think of it like your furnace filter. The side that's to the inside (closer to the furnace) has some sort of a screen/mesh/whatever on that side to prevent the soft side (that collects the dust) from being sucked into the furnace.