We are not to be hypocritical. When we've removed the beam from our own eye, then we can remove the splinter in our brothers. (Matthew 7:5) We can and should judge "righteous judgement". (John 7:24)
The John 8 story is often misunderstood. The woman was alegedy guilty of adultery, but the law had to be followed to the letter. See Leviticus 20:10ff. Both the adulterer and adulteress are to be put to death. Since she was caught in the very act (John 8:3), where was the man? Where were the accusers? Were they sinless IN THIS MATTER? (Obviously not, since the man was absent and the witnesses were caught in criminal misconduct which would have resulted in their stoning--such is the penalty for bring a false charge.)
That has always been my view of the text.