Akatsuki no Yona, Cross Ange: Tenshi to Ryuu no Rondo Recommendations
Princess Ange and Princess Yona worlds turned upside down. Their kingdom back stabs them. Now on the run from them. Both princes hope to change and change the people around them. One series is a mecha anime by Sunrise. The other a fantasy/romance adventure series.
Both revolve around a princess who initially refuses to adjust to her new life she's been forced into, but eventually, come to grips with it, and accept it. Build up a toughness for the real world. They also cut their hair.