Andrew Fishburn is my 4 times great grandfather on my fathers maternal line. My fathers mother was Madelaine Fishburn. I too have recently been doing research to try to find information relating to Andrew's pre First Fleet life, but with the same limited success as others.
A couple of points:
1. Contrary to information given to me by a researcher at the National Archives in Kew, i.e. "All I can say is that a Private in the Marines is very unlikely to have worked in that capacity (carpenter) whilst serving as a Marine." The book 'The Voyage of Governor Phillip to Botany Bay' certainly indicates that there were a number of artificers, including carpenters, masons, etc on the First Fleet, including Andrew Fishburn. Apparently many of the First Fleet marines were former tradesmen displaced by the Industrial Revolution, and their skills helped to build the colony.
2. I have also been in touch with the Whitby Museum and they cast doubt on the veracity that Andrew was related to Thomas Fishburn of ship building fame and particularly that he was Thomas's son. That's not to say that he couldn't have come from Whitby, but no one has been able to provide any evidence to this fact. I suspect too that the claim that he was born around 1760 is based on his approximate age when he died.
I certainly look forward to any further information that may be available.
Don