Is it permissible to marry a girl whose fiancé is said to use vulgar language, as do her mother and sisters, and what is the solution if this is confirmed?

Religion and character are the basis for accepting a woman as a wife. The fundamental principle concerning a Muslim is their integrity until proven otherwise. Therefore, inquire about the girl from someone you trust, and do not break off the engagement based solely on rumors. If what has been reported is confirmed to be true, then if you are capable of reforming her, rely on Allah. Otherwise, seek someone else. You cannot test her directly as she is still a stranger [non-mahram (definition)]. However, you can verify information through trustworthy women, and advise her in the presence of one of her mahrams if you do not fear a greater harm.