What should I do about discovering sexual conversations between my sister and my married cousin on her phone, given my fear of informing my parents or confronting her, considering I learned of this through spying?

You must repent for what you did by spying on your sister. Then you can use an indirect approach to advise your sister and remind her of the danger of Satan and the importance of repentance. If she does not understand or persists, then you must directly tell her that you know about the inappropriate nature of her relationship with her maternal cousin, and call her to repentance. If she does not repent, threaten her by telling your father. If she still does not desist, then inform your father so he can put an end to it, because he is her guardian and responsible for her. And know that it is never permissible for you to kill her, because killing a soul that Allah has forbidden is one of the major sins.