Actually you can use either "lei dice che vuole cambiare" or "lei dice di voler(e) cambiare", but there is a difference between them: the former means both "she says she wants to change" and "she says that he/she ("she" as another woman, not the one who is speaking) wants to change", while the latter only means "she says she wants to change".
Everything is correct with both your sentences except "would like"
"Vuole" means "he/she/it wants"
"He/she/it Would like" is "vorrebbe"
"Would like" is a different tense, so to speak. Specifically it is the present conditional (please correct me if I'm wrong but I'm confident I am correct)