What is the ruling of Islamic law regarding marrying the girl who is currently engaged after she committed to wearing the hijab, or the religious paternal female cousin, especially after performing Istikhara and the vision not being clear, and the father's objection to breaking off the engagement?

It is permissible to break off an engagement, as it is not a binding contract. However, it should not be done without a legitimate Shar'i (Islamic legal) excuse, due to the breach of promise and harm it entails. Obedience to the father is not obligatory regarding marriage to a specific woman. As for consultation, it is advisable to obey the father due to the goodness found in obeying and honoring him, and because he has more experience, and perhaps his choice is based on knowledge. The evidence for the soundness of his choice is the girl's obedience to you, her humility, gentleness, and affection; as per the Hadith (definition): "Marry the fertile, affectionate one." He should frequently perform Istikhara (prayer for guidance) and supplicate to Allah to destine what is good for him.