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.
Not just flannel because that shit gets hot in the summer. There are pearl snap western cut shirts that you can wear in the summer that won't kill you. They're much thinner.
Otherwise, pretty good advice here. And Sam wore those western shirts in addition to flannel.
In most places, hats and coats are considered outerwear. You suggest someone wear a hat, and they'll wear it everywhere.
I suggested ditching the hat (and coat) because people who dress for a theme, like OP, often let the theme override indoor/outdoor considerations.
Maybe in Texas they wear the hats indoors. But in most places, you go inside, you take your hat off immediately (and hang it up until you go back out).