This website will not let me cut and paste, but if you scroll down to the middle of the page, you can read the text of an historical marker that gives a clear and concise summary of the Alabama-Coushatta Indians in the War. They did indeed serve exclusively in Texas, except for some training in Arkansas and a bried stint in the CS Navy in Galveston Bay.
Their reservation is between Tyler and Polk Counties, close to where my mother's ancestors lived from 1838 to the present day. It is one of only two Indian reservations in the state. The Alabama-Coushattas are just about the only native tribe that never fought with Texians. God bless them.