Mr. O’Keefe and his lawyer, Paul Calli, revealed new details about the diary investigation and F.B.I. search to Sean Hannity on Fox News on Monday. During the interview, Mr. Calli said that Project Veritas had paid for the right to publish the diary but was unable to confirm it belonged to Ms. Biden and ultimately decided not to go ahead with a story about its contents. Excerpts from the diary were later published by another conservative website.