What can I do if my face keeps drying and flaking on my upper lip, chin, and eye brows?

Other OTC options. Other over the counter shampoos that you could use as a face wash include head and shoulders (zinc pyrithione) and Nizoral ad (ketoconazole). You can also spot treat areas with 0.5-1% Hydrocortisone cream (e.g.: cortaid). If these don't work see a dermatologist. There are several other prescription topical creams and shampoos available for seborrheic dermatitis.
Selenium. You may have seborrheic dermatitis, in which case, shampooing with a selnium shampoo, such as selsun blue may help your flaking.