What is the way to deal with family members who commit transgressions, such as a sister's immodest display and lack of hijab with everyone's approval, given an inner aversion to these transgressions, and is this aversion natural and justified?

Your eagerness for the righteousness of your family and to change the evil among them is a sign of goodness. Strive to advise them, enjoin good upon them, and forbid evil from them with gentleness and respect, especially with your parents. It is not permissible for you to condone your sisters' display of adornment (tabarruj) in front of non-mahram men. Rather, this evil must be prevented to the best of one's ability, for the Prophet, peace and blessings be upon him, said: "Whoever among you sees an evil, let him change it with his hand; if he cannot, then with his tongue; if he cannot, then with his heart, and that is the weakest of faith." A believer should have mercy on sinners and hope for their guidance, while disliking the evil and hating the disobedience itself.