Adam Levine Hanging with Wife & Kids After Vegas Residency Shout-Out

Adam Levine is a man of his word -- in the wake of his sexting scandal, he publicly declared he was fully committed to his family ... and here's video showing exactly that.

TMZ obtained footage of the singer out doing a little shopping Sunday with his brood, hitting up the H&M store at the Forum Shops at Caesars in Las Vegas ... and rolling in as a tight unit. Both of his kiddos, Gio and Dusty, were there ... as was his wife, Behati Prinsloo.

Watch for yourself ... you can see Adam and co. cruising up the escalator, and when they reach the top -- Adam lifts his son off his shoulders as their daughter runs around near him. Seems they had a bodyguard with them too ... but it was a straight up Levine family outing.

The reason this is notable now ... on Friday -- during Maroon 5's debut concert for their new residency -- AL specifically shouted out Behati and his children during his performance.

During the sweet moment, Adam spoke candidly ... saying he used to be out here (onstage, it seems) for himself -- but at this point, he says he's out here for them. He also said his family is the only thing that truly matters to him in the whole world. Adam said he loved his fam, and looked to his left ... so it appears Behati and the kids were actually there.

The backdrop of all this, of course, is the fact that Adam was accused of cheating last year -- or, at the very least, emotionally cheating ... by allegedly DM'ing a bunch of women.

He denied getting involved with any of the women who claimed he was messaging them, but acknowledged he'd crossed a line and screwed up. It was a whole scandal for a while.

In the aftermath of that, Adam and Behati appear to have reconciled ... as they've been spotted out together and looking happy several different times now -- including here. Also, they just welcomed a third baby into their lives ... although, there's no sign of it in this clip.

Bottom line ... Adam seems to have a new outlook on life, and the crowd at his first show appreciated it -- giving a good round of applause, with cheers to boot.