Yup, your grandmother was correct for the most part.
William Coats of Texas County early in the war was a captain in Company D of Colonel William O. Coleman's southern regiment from this area around Dent, Texas, Shannon, and other nearby counties. This is according to Broadfoot's 1990s "Supplement to the 'O.R.'" part 2, vol. 38, on page 270. Coleman's regiment raided Union wagon trains and patrols in this region throughout the war. They were "regular" soldiers properly mustered in and answerable to Confederate authority, but they fought as "irregulars" or guerrillas or bushwhackers against the Union troops who occupied Missouri.
Joanne Chiles Eakin's 1995 list of MO POWs from the St. Louis area prisons did list a "Private William T. Coats," but the only detail remaining on the Union military prison record was that he was released from the Alton, IL military prison after taking an oath. This may or may not be the William Coats from Texas County.
A Union militia lieutenant, William L. Boyd, (in "Official Records" series 1, vol 34, part 1, p. 115) on 18 Feb 1864 mentioned your Bill Coats, too. He put in his report that he heard "the the guerrilla Bill Coats had been the night before at a house on Hamilton Creek [runs just east of Morris Township a few miles southwest of Houston]. We learned that Coats had moved his camp to Big Piney."
I looked up Bill Coats in the online MO State Archives on the MO Sec'y of State's website and found only the mention that Private W. M. (or William) Coats of Company C, Coleman's regiment, attended the 6th Annual Reunion of the MO UCV (United Confederate Veterans, I think) at Houston, MO.
I think I found Bill Coats in the 1860 census of Morris Township, west-central Texas. It lists William Coates, age 30, born TN, a farmer, whose real property he valued at $480 and personal property he valued at $425 (unless I got those backwards). His wife Rebecca W. Coates, age 32 was born in KY. Their four young children, aged in 1860 from one to seven years were all born in MO.
I realize these are bits and pieces, but it does prove that your grandmother was correct. Now, aren't you ashamed for doubting her? Your Granny would have wanted me to say that.