Is the woman who cut ties with her aunt, who insults her in front of people and refuses to shake her hand, sinful, or has she fulfilled her obligation?

It is recommended to supplicate for the mending of the relationship with your maternal aunt, and to seek the reason for her behavior in order to correct it, and to strive for reconciliation through all possible means. Indeed, mending discord between people is one of the best acts of worship. If the situation does not improve, it is not permissible to sever ties with her; rather, you must maintain connection with her through whatever means of kinship are available, while being sure to offer her greetings of peace even if she does not return them. You are not obligated to shake her hand. It is recommended to visit her, even if you are alone, because maintaining kinship ties is obligatory, even if met with estrangement and mistreatment.