Does divorce by Khula' take effect if the husband does not explicitly utter the words "you are divorced," and does this affect the validity of Khula'?

If your husband separated from you through divorce or khul' before consummation, you have become irrevocably separated from him. It is not a condition to hear the pronouncement of divorce or khul'. You may marry someone else. Avoid obsessive thoughts and turn away from them. Seek Allah's help and occupy your time with what benefits you.