Is there any other history of the travel of the original document? I know that Monroe Madison saved it from being burned during the war of 1812, and it went around the United States in the 1970s.
Edit: in trying to find an answer for this, I found this Wikipedia page, which is very relevant and useful.
There are several different copies. The first copy is called the "fair copy" and no longer exists and may have been purposely destroyed the night it was adopted by Congress, on July 4, 1776. It was taken to a local printer that night, who transcribed it into typeface, and printed out 200 copies. These are called the "Dunlap broadsides" and a number of them still exist. It is one of these that was sent to George III. I'm not sure if that particular copy still exists.
Another Dunlap broadside was transported from Philadelphia to New York and given to George Washington who read it to his troops the night of July 9th. That copy still exists in the Library of Congress.
The one signed by John Hancock and all the members of the Continental Congress is called the "engrossed copy" and wasn't prepared until August 2. That one is the one that was preserved by Dolly Madison, according to legend, and reproductions of it were made beginning in the 1820s as the original began to deteriorate.
So, the popular legend is untrue. Hancock didn't write his name so big for the king to see it, because he never saw it. The copy sent to the king was in typeface, and had been sent weeks before Hancock penned his famous signature.
Are you positive Monroe saved it in 1812? Madison was President that year.