Jefferson chose to begin the Declaration of Independence with the words, The unanimous declaration of the thirteen United States of America. Was this necessary? could the 13 colonies have declared independence if they were not unanimous?

Yes, it was necessary. The reason why Jefferson made such an introduction was to provide a clear statement that the initiative of independence was a cause common to all the colonies, therefore, it would project a stronger and more concise resolution for such an intrepid decision. That's why they opted to do it as the United States of America, to show that they were all in for the cause, instead of doing it separately or not showing that it was a unanimous decision.