Hello and welcome to YT! so glad you and Princess found this site! It's so full of great info and wonderful friends!
Princess could be alergic to her food, Science Diet has corn in it and some dogs are alergic to corm, I would try to change her food and make sure you pay attention to the ingredients, there is a thread about this and I advice you to read it.
http://www.yorkietalk.com/forums/showthread.php?t=48516
also if I were you I would use a good dog conditioner on her, that will help with her dry itchy skin.
Good luck to you and I look forward to getting to know you and Princess better!