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.
Check out Sam from TrueBlood as a good example.
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.
If you want what real current day cowboys wear.
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Boots (no over the top clasps, spurs, and stuff. Go for something like this: https://www.justinboots.com/)
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Light blue jeans pants
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white t shirt, can substitute a linen or seersucker button down if preferred. Plain or checkered pattern if doing button down.
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western hat
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belt with big buckle (optional)
Source of my authority on this matter: Lived in Texas since I was like 10 or smth
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).