I will tell you my experience. I got diagnosed about 4 years ago. I have had this since I was at least 16. It started on my head, and when I was 18 I had it on my face. Man, that was horrible. It was red and it hurt it was so bad. I'm 44 now, so I've had it forever...doctoring myself with hydrocortisone and stuff. ..shudder. Anyhow, I will tell you what worked for me. I experimented. What works for me may not work for you.
When I was diagnosed, I had a prescription shampoo...Nizoral or something. I also had a cream for my face...two actually...a steroid and another anti-fungal I think. I had to use those twice to three times a day for the rest of my life the dermatologist said. I found "naturally curly" after that. I ditched the shampoo and started co washing. The tea tree conditioners are fantastic. They helped the itching. Around the same time, I started doing the oil cleanse method that I saw on the makeup/skincare boards. Now, I'm using coconut oil to cleanse and moisturize my face. So, oils are definitely helping me. I also put it on my scalp at night. My SD at the moment is under control.
There was also a thread on here about using honey on your scalp and face. That helped. If I were you, I would experiment. Sometimes I have no faith in what doctors tell me I have to do. They really push the steroids. My husband's doctor advised oral steroids for his skin, and he had a major skin rebellion. He broke out everywhere. He has a different type of eczema.
Anyhow, that's what worked for me. I also take an omega 3 and black currant oil. The coconut oil really helps though. Olive oil was nice, but the coconut oil was the best. Rosemary seems to help as well. It's in my trader Joe's conditioner. Google SD and read all you can. Try stuff out. Green tea rinses help as well. Also, apple cider vinegar rinses help. I experiment. :-) I cut out a lot of dairy too...not sure if that helps, but something is working. Oils and conditioners are my best friends. :-). I do have a lot of shedding. I don't know if that is from the SD or not. I did read that it can cause hair loss.
Hope some of this helps.