Can a person who finds out that someone is a sex offender go around telling everyone that that person is a sex offender?

The issue here is one of privacy rights.  How the state of Texas handles sex offender information matters.  Many states, such as New York and Alaska, have on-line registries.  There is, therefore, no expectation of privacy as to the information.  Massachusetts requires that law enforcement notify community groups and neighbors of "moderate" or "high risk" offenders in the area.  It is my understanding that Texas maintains a public database of registered sex ofenders so I am going to take the position here that there is no cause of action for "verbal assault" - do you mean defamation? truth is always a defense here - or harassment.  But check with an attorney in your area to be sure.  

There is no such thing as "verbal assault." "Harassment" is something done to the target--saying things about the target to other people is not harassment. If the statement(s) were untrue, then this might be defamation: the public (even to one third-party or other person) making of false statements of fact that damages a person's reputation and/or makes others less likely to do business with that person. However, the truth is a defense to a claim of defamation: that is, if the statement(s) the person is making are true, there is no defamation and no cause of action.

About the only time relating true information about one person to others might be actionable is if that information was private, so that relating it was a violation of privacy; that would not apply to information that was part of the public record.