Is the questioner considered sinful or disobedient by refraining from caring for her sick mother, given her mother's history of illicit relationships and repeated attempts to involve her daughters in them, and the resulting deep psychological harm to the questioner and her sisters?

The mother's right is great and is not forfeited by her immorality or transgression. Allah has commanded to accompany disbelieving parents with kindness. Her rights include enjoining good upon her and forbidding evil from her, caring for her, and serving her, especially during her illness and weakness. Serving her does not necessitate living with her, but it is not permissible to abandon her.