This man was in service of "those people", so his military records have no connection to Alabama. However, our records service (available through the message board - see above) should be able to access them from Washington DC.
The last name usually appears in the records as Winrich, Windrich, Wenrich, or Wendrich. This is a common name in Pennsylvania, where it's usually spelled Wenrich or Wenrick. David and Frederick Winrich appear on the 1860 census of Dauphin County PA, post office Harrisburg. This is probably the same man as the one who enlisted in Federal military service.
During the war DW served with Co B, 15th U S Infantry Regt as a wagoner. He later (post 1865?) appears on roll with the newly formed 33rd U S Infantry Regt, also as a wagoner with Co B. From late September of 1864 through July 1865 the 15th U S Infantry Regt was stationed at Chattanooga, within easy distance of Jackson County. We might assume that he received a discharge before 1870 and settled in the area.
Both service records shold be useful to you.