I'm sorry to hear about your situation. First of all, this probably is not defamation (slander is spoken defamation, libel is written defamation... IF it were defamation, it'd be libel). The reason is that defamation is a specific legal definition that consists of a false statement of material fact, communicated to a third party, that causes quantifiable damages. If this site that published this information was publishing false information, you could potentially have a case. But if it's true, or they're only publishing what's "public record" (which is likely where they got this from) they wouldn't be publishing false information, as they wouldn't be making a claim to the truth of the matters contained therein, but rather only to the existence of these criminal records. Thus defamation almost certainly wouldn't work.
"Distress" is not a legal cause of action. It could be a justification (an affirmative defense) if you were to do something and they were to sue you, or some criminal prosecution were instituted against you. It's typically the damage that you suffer (emotional distress, pain and suffering) that "distress" refers to. But one can suffer damages without having a case against anyone. Damages are a necessary element of a cause of action, but not sufficient. That is, you have to have the other elements of the cause of action (in a defamation action it would be a false statement of material fact to a third party) as well as the damages (which emotional distress could play a part) to be able to win. If you can't prove all of the elements, you wouldn't win.
Unfortunately as these are public records, they can be sold by private entities, or hold you "ransom" to remove the information. It's absolutely unfair, unethical, and immoral, but it is legal. And even though there are efforts to outlaw such actions, it's unlikely that such laws would survive any constitutional challenge, as they would be infringement of speech.
I know this is probably not what you wanted to hear, but it is the law. I hope that clears things up anyway. If you have any other questions, please let me know. If not, and you have not yet, please rate my answer AND press the "submit" button, if applicable. Please note that I don't get any credit for the time and effort that I spent on this answer unless and until you rate it positively (3 or more stars). Look for the stars on your screen (★★★★★). Thank you, ***** ***** luck to you!