“Jesus didn’t condemn the woman caught in the act of adultery, but condemned those who judged her. Therefore you shouldn’t judge others.”
The Christian is not “judging others” but simply telling the world of God’s judgment—that God (not the Christian) has judged all the world as being guilty before Him (see Romans 3:19,23). Jesus was able to offer that woman forgiveness for her sin, because He was on His way to die on the cross for her. She acknowledged Him as “Lord,” but He still told her, “Go, and sin no more.” If she didn’t repent, she would perish.