A new filing in a California court portrays Trump attorney Michael Cohen in action as the president's fixer-in-chief. Adult film actress Stephanie Clifford, whose stage name is Stormy Daniels, has filed a lawsuit against her former attorney, Keith Davidson, who she claims colluded with Cohen to cover up her affair with Mr. Trump.

The filling by Clifford's current lawyer, Michael Avenatti, quotes text messages between Davidson and Cohen about booking Clifford on Fox News' "Hannity" after a January 2018 article in In Touch magazine that detailed the claims of the affair. Avenatti told BuzzFeed that he obtained the text messages from Davidson after "months of demands."

"I have her tentatively scheduled for Hannity tonight," Cohen wrote to Davidson in one text. There are no texts that explain the reason for booking her on "Hannity," but the brief claims "on information and belief" that Davidson and Cohen "hatched a plan" to have Clifford appear on the show "to falsely deny the accuracy of the In Touch article." It also claims that Cohen tried to put her on the show "only after consultation with Mr. Trump."

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But Clifford was flying to Los Angeles that evening and wasn't able to make the appearance. "This is no good," Cohen said in another text. "We need her as by doing tomorrow you just create another news cycle instead of putting an end to this one." About 30 minutes later, Cohen backed off of that plan. "Let's forget tonight. They [Fox News and the Trump administration] would rather tomorrow so they can promote the heck out of the show," he wrote.

Then, two hours later, another change: "The wise men all believe the story is dying and don't think it's smart for her to do any interviews. Let her do the thing but no interviews at all with anyone." "The wise men," the filing claims "included Mr. Trump." It doesn't say what "the thing" might be.

Avenatti argues that the appearance was meant to "provide a false interview and lie to the American people to serve the best interests of Mr. Trump and Mr. Cohen."

Clifford's lawsuit also accuses Davidson of tipping off Cohen to the lawsuit she planned to file against Cohen and Mr. Trump, which would disclose her prior affair with Mr. Trump.

And the brief also claims that Cohen then tried to anticipate and influence first lady Melania Trump's reaction to the impending lawsuit by Clifford.

"After learning of Ms. Clifford's plans through the improper disclosure of confidential information by Mr. Davidson, Mr. Cohen undertook efforts to meet the next day with Mrs. Melania Trump, in order to 'get out in front' of the approaching lawsuit and publicity, and convince her that Ms. Clifford was a liar and not to be trusted."

The brief quotes Davidson asking Cohen to call him -- "U calling?" And Cohen responds, "With flotus. Give me a minute."

Cohen apparently recorded conversations with Davidson, and Clifford and Avenatti have sought those recordings, which they say include attorney-client privileged information. They say he has refused to provide the files. Clifford is suing for her client file from Davidson, as well as any text messages and correspondence that relate to her.

Brent Blakely, Cohen's attorney in California, said that Clifford's suit was without merit. "This new lawsuit filed by Stephanie Clifford aka Stormy Daniels has no merit whatsoever as to my client, Michael Cohen, and appears to be yet another publicity stunt. We look forward to defeating this lawsuit in court," Blakley said in a statement to CBS News.

Davidson also released a statement in response to the lawsuit: "This outrageously frivolous lawsuit is yet another desperate attempt by Michael Avenatti to continue his 'publicity tour,' as well as divert attention from the recent allegations against him relating to bankruptcy proceedings and the failure to withhold millions of federal employee taxes. That said, Attorney Davidson is very happy that he has filed this lawsuit because he strongly believes that the filing constitutes a full and complete waiver of the attorney-client privilege."

Davidson has been accused of similar dealings involving a former Playboy Playmate, Karen McDougal, who also alleges she had an affair with Mr. Trump. Before the 2016 elections, she signed a deal with American Media, Inc. (AMI) to squash the story, a practice known as "catch and kill." McDougal filed a lawsuit in March to nullify the agreement.

McDougal's lawsuit claims AMI worked secretly with Mr. Trump's associates and Davidson, then her lawyer, to bury her story about an extramarital affair McDougal says she had with the president. She claims that Davidson was a close ally of Mr. Trump and was only pretending to advocate on her behalf, and it also claims that Cohen was secretly involved in this effort.

CBS News' Laura Strickler and Clare Hymes contributed to this report.