Remember that it takes several times longer to typeset a page (by hand, as in 1776) than to hand-write it; and that the typesetter still requires a hand-written fair copy to set from. So you don't save any time by only typesetting the document - as it must be first written out fair for the typesetter.
From Wikipedia on the Dunlap Broadside (my emphasis).
On July 4, 1776, Congress ordered the same committee charged with writing the document to "superintend and correct the press", that is, supervise the printing. Dunlap, an Irish immigrant then 29 years old, was tasked with the job; he apparently spent much of the night of July 4 setting type, correcting it, and running off the broadside sheets.
In the terminology of the time, a fair copy was the (error and correction free) copy made, for distribution, after all drafts were complete.
In high school I belonged to the club that volunteered to hand-set and print the school's brochures, flyers, and event programs on an ancient hand press. Modest experience only; but based on that I estimate that an experienced typesetter was no more than only a fifth or tenth as fast as a good calligrapher in producing a document. While multiple pages could be set in parallel by multiple setters, that is difficult to do for a single page.
My guess is that a first fair copy was penned by Jefferson from the drafts for signing, and then either it or a second was supplied to the typesetters for working from.