I'd guess Cinderella. While she will become a princess when she eventually marries the Prince, she is not one throughout the show. The trio are the step-mother and step-sisters, it's an R&H show recently on Broadway with a largely female cast that's family friendly.
While it doesn't perfectly fit all of the criteria, Mary Poppins could be a possibility. It just came out for licensing and it very popular with high schools. In fact, my hs did the show and it went wonderfully! I do also think it could be The King and I.