I posted this on the La. board in response to your query there, but am posting it again here just in case you don't find it there.

Unless there was more than one soldier with this rather unique name, I found him in two other commands (did not find him in any La. unit):

Ulysses A. D. Weathersby
Morgan's Reg't Texas, Company A
Private

Ulysses A. D. Weathersby
Baylor's Reg't Texas Cavalry, Company H
2nd Lieutenant, promoted to 1st Lieutenant

Looks like he may have migrated to Texas *during* the war, or got hooked up with Texas boys during the war and they convinced him to move to Texas after the war.