Dear Mr. Winn,
Thank you for posting your request on History Hub!
The transcription of the Association of Members of the Late House of Burgesses [27 May 1774] available on Founders Online that is taken from the Papers of Thomas Jefferson at Princeton University, cites a broadside copy at the Library of Congress and also states that there is a copy at the New York Public Library. It does not mention a surviving handwritten copy of the document. We suggest that you contact the Library of Virginia as they are the official record keeper for the Commonwealth of Virginia and may have additional information about records of the House of Burgesses.
We hope this is helpful. Best of luck with your research!