I don’t know any soaps. But when my son had it around 2-3 months old, I’d just use the hospital brush to remove the flakes and to this day I still put a little baby lotion in his hair 🤷🏻♀️👍🏻
Try to do pure coconut oil and or olive oil massage rinse and get the cradle cap brush usually less than $10 at target or Walmart brush and let it be but if you want to do something for the dry get some coconut oil or nature oils