You can do the cowboy look well. Ditch the long leather coat, the hat and the toothpick. Wear flannel button-ups and cowboy boots of a standard color with jeans of a standard color.
Don’t copy phrases from movies, but speak slowly and politely. Don’t do a strong accent, but it’s ok to do a mild accent. Instead of tipping a hat, give a significant, thoughtful head nod as a greeting.