The problem-areas and symptoms you described are usually associated with seb derm induced by malassezia. Cetaphil will make malassezia worse, but because you have dry skin, you could use dabs of MCT oil, which is sold in the Sports Nutrition section of Whole Foods, or many places on the web. MCT oil isn't going to cure anything, but it won't make malassezia worse while moisturizing.
Lotrimin Ultra (not the AF kind) is the only effective OTC anti-malassezia cream treatment, and it's sold in every drugstore.
Also, make a web order for an anti-fungal like Nizoral 1% shampoo (ketoconazole) or Hegor 1.5% (climbazole) shampoo (check eBay) and use this as face wash and shampoo. These are not sold in stores but they are non-prescription. After using one or the other of these for 3-4 days, stand in bright sunlight and scalp your scalp. If you see a small cloud of dandruff flakes, then you have a confirmation of seb derm induced by malassezia. It's like dandruff but more adherent, and often invisible until you use an anti-fungal. Selsun Blue1% (selenium disulfide) will evoke the same response, but is not effective after the first week. Head and Shoulders (pyrithione olamine) is a complete waste of time.
The condition is easiest to understand as a skin allergy, evoked by the oil-eating metabolism of malassezia, but it's more complex than that. Look for anything I've posted on this forum for more information.
haha, oops --"scratch your scalp" not scalp your scalp.