When the Obama campaign sent popular, telegenic, man-of-action Newark Mayor Cory Booker out to do a little surrogate-advocacy on their behalf last month, something happened that the Democrats certainly did not intend: Cory Booker made a very public display of intellectual integrity (I know, I know — a rare beast in politics, but it can happen). On “Meet the Press,” Booker said that Team Obama’s attacks on Mitt Romney’s record at private-equity firm Bain Capital and other such petty campaign tactics were “nauseating” and made him “very uncomfortable.” And it felt so good.

But the Democrats didn’t take kindly to Booker making such a distasteful spectacle of himself and shooting one of their most prominent attack memes straight through the heart, and before the day was out, Booker had released a four-minute explanation that even the MSNBC anchors called a “hostage video” (heh). Then, Booker went on Maddow’s MSNBC show and made it look like he had been commanded to walk back his comments from on high — and the hits just kept on coming.

Booker is a relatively young and clearly pretty ambitious politician — but he’s going to paying for this one for a long time yet.

Newark’s mayor, who was gunning for a spot in President Obama’s Cabinet, lost the chance after he shot his mouth off during a blunderingly honest TV appearance last month, sources told The Post.

“He’s dead to us,” one ranking administration official said of the prevailing feelings at the White House and Obama headquarters in Chicago.

Booker had been angling for the housing secretary gig in a second Obama term, according to sources in the administration and close to the mayor. …

“Cory and Barack Obama have never been besties, but that was the final nail. It’s like, ‘You’re dead and done.’ The firing squad is out,” said a Democratic source in contact with both sides. “It’s not just that he messed up, it was that he compounded it and didn’t have their back when they gave him a national stage to talk.”

The Obama camp tried to play down the bad blood with Booker yesterday.

“Mayor Booker’s comments are behind us and we are working together to re-elect the president in November,” said Patrick Gaspard, head of the Democratic National Committee.

Other campaign insiders insist that Booker is not persona non grata, although he is in the doghouse and knows it.

“What he did was undermine a leading argument for the campaign. It was a serious distraction. Not a minor screw-up; a major screw-up,” a Democratic source said. “He is trying to figure out how to work his way back in the fold.”

Last weekend, Booker did a West Coast swing for the Obama campaign and is already scheduled for upcoming appearances.

In Seattle, Booker joked that the campaign had instructed him not to “stick your foot in your mouth again . . .So this is my probation speech.”

When so many of Obama’s faithful minions are so awkwardly wandering off-message, you’d think they’d stop and maybe reassess the situation. “Hmm, maybe the problem isn’t our meandering surrogates… maybe the problem is us and our free-enterprise hatin’, class warfare-mongering, intellectually cheap campaign-rhetoric machine?” I won’t hold my breath.