Good luck on your search. You appear to be on your way. The following may not help but you can never tell:
The "Shaw Neighborhood" in Washington, DC is named after Col Shaw and the plaque that honored him at the Boston Commons was moved there several years ago. The monument to ALL black civil war soldiers is also there and Col Shaw is listed on it. You might gather more info on him by "visiting" the Webpage associated with it. White officers are listed right along with the African-Americans.
Some years ago, the Clan Shaw Society (a Scots heritage group) interviewed a nephew of Col Shaw -- perhaps it was your grandfather -- and donated a book to the Shaw library in Washington. I have a copy of the book somewhere in my library but cannot recall the name at the moment.
My name is Shaw and I was told I was descended from an officer who led a black unit in the civil war and when I heard about RGS I naturally assumed he was the one. But as I became interested in genealogy I learned all my Shaws came from the south. More research revealed that my Major Orrin M. Gross (also listed on the monument above) led a U.S. C. T. artillery unit.
If you have specific questions, I suggest you refer to the sources above and the genealogist for Clan Shaw Society. I will also answer any questions I can, but I know less than the people above.