Yes, Washington W. Lee did serve with the "Pulaski Blues", Company F, 31st Georgia Volunteer Infantry.
In my roster I have that he enlisted Oct. 27, 1861, as a private. He died of typhoid fever at Chimborazo Hospital #3, Richmond, VA, on July 14, 1862. Buried at Oakwood Cemetery, in Richmond.
Do you have a photo of him in uniform, or copies of any of his war letters?