Hmm i'm asking my girlfriend about your question since she is the real shampoo expert in this house :) The Coconut and egg based shampoo is for dry hair but for a dry scalp she suggest you use ''Tea Tree Oil'' i've read about it before this oil disinfects and helps against itching. Add a few drops to your shampoo and conditioner. Make sure that the oil does not come into contact with your eyes. Basically oiling is the best option to fight against itchy and dry scalp, but if tee tree oil doesn't do the trick you can also use argan oil, eucalyptus oil, jojoba oil, peppermint Oil .. But they should be applied cold on your head. Massage a few drops of oil after washing your hair. Repeat treatment during the week if necessary. Only use oil on a cleaned scalp and not on a layer of flakes. This can make the situation worse. Apple Cider Vinegar is also a good choice in neutralizing the pH levels of your skin.
A quick google search also provided me with this recipe based on coconut oil (and yes i'm a fan of this product since i use it in almost everything) But the recipe goes like this:

1.Warm a cup of coconut oil and then add a few drops of lemon juice. The lemon juice is good because it removes excessive dead cells.
2.Apply to your scalp and leave for at least an hour.
3.Wrap your hair in a warm towel for a while.
4.Remove and wash your hair.

I hope this information is usefull for you and will help you further with your problem goodluck @karenmckersie