I have a copy of Terrell's Texas Cavalry by John Spencer. There is no one named Washington on the regiment's roster in company "D" or any other company. According to page 110, the roster listed in the book was taken from the only roster to survive dated February 29, 1864. The regiment was organized in July 1863. According to the footnote, the names are shown on Micro Copy No. 323, Rolls No. 177, 178, and 179.
You referenced the 11th Tx Cavalry. The company "D" commander was George Washington Diamond. He had been in the 11th Tx Cavalry for a period, but in the spring of 1863 he enlisted a new company along the lower Brazos River which became Company "D" of the 37th Tx Cavalry. I checked the 11 Tx Cavalry, but there is no Washington listed in FOLD3.