Sawrey was born in 1765, received his professional education from Andrew Marshal, M.D.
(1742–1813), who taught anatomy privately in Bartlett's Court, Thavies' Inn, from 1785 to 1800.
Sawrey attended Marshal's lectures in 1794, and attracted the attention of his master by a dissection of the nerves of the eye.
He was admitted a member of the Corporation of Surgeons on 7 July 1796, and he acted for some years as demonstrator to Marshal.
He practised his profession in both places, and in later life turned his attention more particularly to ophthalmic surgery.