Are there any cases where someone disclosed a trade secret but did not have the third party sign an nda ...

You could try to argue that you had an oral agreement with the other party to keep the information confidential. Oral agreements are very hard to enforce if there is no written evidence, and there are other obstacles such as the Statute of Frauds that prevents certain oral agreements from being enforced. The best way to protect yourself is through a written agreement signed by both parties in advance. Depending on the idea, though, if you are the author or inventor, you could register the copyright or file to patent the invention. Only the author (in the case of copyright) or the inventor (in the case of inventions) can file for copyright or patent protection so you could shut them down through these avenues if they apply. You should talk to an intellectual property lawyer about the next steps to take.