Wow, I hate to disagree with what sounds like a sterling memory but I just viewed this film (for the second time) in its entirety and the woman who sings in the strip club (a song called "Liz") is definitely not Ann Richards (although Richards does appear in the film.) I believe the woman who sings in the club may be a woman named Gaye Merritt who's name appears in the film's credits not only as having appeared in the film but also having written both the music and lyrics to "Liz".
Ann Richards does have a role though. She is definitely heard (but not seen) singing "Angel's Flight" over the opening and closing credits of the film and she is seen seated at the bar next to the male lead character in that same strip club where the other girl sings (Richards appears annoyed with the male lead character and gets up and walks away from him.) Ironically that scene appears shortly before the singing scene and the singing scene is hazy and shot from a distance so it might be easy to confuse things and assume Ann Richards not only sang the opening and closing theme of the movie but also sang on stage in the club. But clearly she did not, if you compare Ann Richards' voice, heard clearly on the theme song "Angel's Flight" to the voice heard in the strip club singing scene the voices are not the same plus the woman on stage is clearly dressed very differently than Ann Richards who was seen seated at the bar just minutes before. Even the hair style and figures are drastically different between these two women. I am 100% percent positive about this, I have studied Ann Richards music very well as I am a huge fan. I'd know her voice anywhere. Maybe at the time the extras in the film were somehow led to believe that the singer on stage was Ann Richards but it wasn't! At least not in the version of the film that was released.