IRVING, Texas—Even on a rare “victory Monday,” the Dallas Cowboys found themselves inching toward controversy. At quarterback, no less.

Cowboys coach Wade Phillips opened the door by hesitating on whether backup Brad Johnson will remain the starter Sunday against the New York Giants in what should be the final game Tony Romo misses because of a broken pinkie finger on his throwing hand.

The only other quarterback on the roster is Brooks Bollinger. He’s started 10 games in his six-year career, nine with the New York Jets in 2005 and one last year for the Minnesota Viking...

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