Hi, a welcome to you and Rosie. Your little girl sounds so sweet and it is always a joy to hear new owners fall in love with their fur babies.
Sometimes if the shampoo isn't rinsed out well from bathing that could cause irritation. Or could be a food allergy, other allergies, some have had to replace thier new carpet as it contained something that caused skin irritation in their Yorkie...There are so many factors that could be taking place.
Fleas maybe? Flea saliva can make a dog go nuts and scratch.
I hope your Rosie gets relief and we hear all kinds of great stories about her.
Feel free to post pictures if you'd like to share.