I suggest that besides using a quality moisturizer, see a doctor and ask for a mild cortisone such as "Elecon" cream or ointment. Use it for short periods. You can even mix it with your moisturizer. It won’t cause any long term damage if you use it sparingly.
I had a dry scalp problem for years, and it has disappeared ever since I went sly. I don’t put the cortisone on my head as I don’t need it there anymore, but I use it on my nose, face and chest when a get a little eczema attack there. One application and it disappears for about a week.