Should I distance myself from my friend after she said, "This does not concern you," and was it wrong to seek advice from someone else to counsel her?

You have done well in advising your friend and forbidding her from evil. Your informing someone who can advise her, after she refused your advice, is not backbiting that is forbidden. Rather, it is among the situations where backbiting is permissible for the purpose of seeking help to change an evil. If your friend repents, do not abandon her. If she continues in her forbidden relationships, it is better to distance yourself from her and associate with righteous women.