First, you need to be sure you are actually losing more hair than the 100 strands per day average which is normal. Is your hair a lot thinner than before? Are your periods regular? These are some of the things you need to explore with your doctor. Generally, dermatitis doesn't cause hair loss. For keeping the hair clean, any mild shampoo will do. I can't make much comment about the increased oiliness of the hair except to say that I can't see any connection with hair loss.
Best.
Dr. Rockoff