I believe this is the same Andrew Morrison that is mentioned in my family history book. If so, he was born in Belfast, Ireland, in 1733, immigrated to America between 1747 and 1750. He married Mary Organ of Orange Co, VA, and settled down there, where they had 10 children.
There are also records of an Andrew Morrison that served in the 3rd VA Regiment in the Revolutionary War. Although his rank is not listed, it does say he died in the Battle of Brandywine 11 Sep 1777.