Tootstl, I believe Buckytom hit the culprit.
Tell me: Are these spots you describe small in size (1/4" and less) and primarily located on lower or interior older leaves? If so, grab a magnifier and look for a fuzzy, cottony growth within these spots. Next step, if you rub the spots, does the white fuzz rub off and is the leaf surface beneath them darkened and mottled? Are the leaves with lots of spots turning yellow? If this is the case, you have a 95% likelihood of having powdery mildew, which is a fungal disease.
If this is not the case, please advise, because it changes everything!
My best advice, based upon my own plant survival tactics this year, is to judiciously clean the area beneath the plant -- remove all old plant material -- and remove as many infected leaves as you can. Routinely spray with Neem Oil or your fungicide of choice. (As mentioned above, I have been, but I still have problems...)
As far as cooking with this oregano is concerned, wash, wash, wash the stuff. As long as you bring the oregano to a boil in a sauce, for example, and retain the boil for a few moments, you wil kill any and all fungus that remains on the leaves, and you will kill any thing else, for that matter. I would suggest that you not use any but the best leaves (unaffected) for uncooked dishes and, even if they are perfect in form, I cannot stress enough the rinsing process!
But you're an organic gardener. You know that!
Pain is inevitable. Suffering is Optional.