If Sessions’ recusal was his original sin, Trump has come to resent him for other reasons, griping to aides and lawmakers that the attorney general doesn’t have the Ivy League pedigree the president prefers, that he can’t stand his Southern accent and that Sessions isn’t a capable defender of the president on television — in part because he “talks like he has marbles in his mouth,” the president has told aides.

Another on-air reporter is leaving Fox News over frustrations with the direction and tone of the network, the second in the last three weeks to defect for those reasons. Adam Housley, a Los Angeles-based reporter who joined Fox in 2001, felt there was diminished opportunity at the network for reporters and disapproved of the tenor of its on-air discussion, according to two former Fox News employees with knowledge of his situation.

Housley believed that as the network’s focus on Trump has grown — and the number of talking-head panels during news shows proliferated — it had become difficult to get hard reporting on air, according to one of those former employees. “He’s not doing the type of journalism he wants to be doing,” the former employee said. “And he is unhappy with the tone of the conversation of the channel.”

Housley’s objections to the Trump-era Fox News are widely shared within the network’s reporting corps, according to current and former employees of the network. Conor Powell, the former Fox News Jerusalem bureau reporter, left the network earlier this month for similar reasons, according to a person close to him. “People are losing their minds,” one current Fox News personality said, adding that reporters have relayed in conversation that the climate for them is worse than ever before.

Donald Trump’s own lawyer, Rudy Giuliani, distanced himself from his client’s suggestion that it “almost ought to be illegal” for “flippers” to get plea deals in exchange for their testimony.

During Mr. Giuliani’s days as a United States attorney, his office was labeled the “House of Pancakes” for the parade of suspects who “flipped” to try to reduce their prison sentences.