ELKO, Nev.—The battle between Sens. Marco Rubio and Ted Cruz intensified as Mr. Rubio tried to cement himself as the clear establishment candidate for the Republican presidential nomination and Mr. Cruz ousted a top spokesman amid allegations of dirty tricks.

Mr. Rubio racked up endorsements from Republican elected officials, taking advantage of Jeb Bush’s departure from the race after a poor showing in Saturday’s South Carolina primary, upping pressure on Mr. Cruz as they both try to tarnish the luster of front-runner Donald...