This is a Library of Congress record of a pardon by James Madison. It was for a slave in the District of Columbia, on a case Madison thought clearly the judgement excessive.
Monroe signs the pardon first with “The Year of Our Lord, One Thousand Eight Hundred and Nineteen” but then “and of the Independence of the United States the forty third.”
This teases me to find more examples of this dating scheme. The most notable one I can think of off hand is the more literary version in the Gettysburg Address (“Four score and seven”.) There must be other examples early in the Founding.