Trump attorney Rudolph W. Giuliani escalated his media blitz Thursday against the president’s twin legal perils, calling on Attorney General Jeff Sessions to shut down the special counsel’s Russia investigation and justifying hush money to porn star Stormy Daniels as an attempt to protect Mr. Trump’s family instead of his campaign.

President Trump, after numerous statements to the contrary, acknowledged Thursday that he had reimbursed personal attorney Michael Cohen for the payments to the adult-film actress, whose real name is Stephanie Clifford. But the president insisted that the October 2016 payoff had nothing to do with his campaign for the White House.