Nikki Haley and Ron DeSantis traded attacks on a debate stage on Wednesday night as Donald J. Trump basked in the applause of a friendly audience on a different television network, dueling appearances that showcased how the Republican primary race increasingly feels like a contest for second place.

At Drake University in Iowa, Mr. DeSantis and Ms. Haley engaged in two hours of verbal combat in which they showed more determination to relentlessly score points against each other than present a cohesive vision for the American people — or attack the dominant front-runner, Mr. Trump.

At times, it seemed as if Mr. DeSantis and Ms. Haley thought the winner would be determined by whoever spoke the most words per minute, racing through attacks at such speed they were all but indecipherable.

With a platform all his own, Mr. Trump appeared to keep marching toward the party’s nomination. He even broke the news of the night — in a way that revealed just how much he’s looking past his supposed rivals — when he said that he had made his decision on a running mate, though he did not give a name.