Draftees were indeed sent to the Marines at that time. At the peak of the manpower shortage, at the induction center those in charge would count out "One, two, three, Marine!" So, one in four draftees (at the height of the conflict)went to the Corps. I later asked a Marine Corps Sgt. if he had any contact with Marines who had been drafted. He said that indeed he had a couple working for him and they were every bit as good as volunteer Marines.
Please don't get me wrong. I don't doubt that Marine draftees were every bit as good as the volunteers. I just didn't want to be one of them. :=)
A friend who had been drafted into the Army told me that his service number (which is used for everything in the military) distinguished him from a volunteer. I recall him telling me that his number started with the letters "US" (indicating a draftee) and volunteers' numbers started with "RA" (for 'regular Army'). I've never independently confirmed this but I believe it to be the case.