When I think of how cowboys talk, at least these days, the voices I hear are Chris LeDoux (actual rodeo cowboy as well as one of the last popular -- as in, non-niche -- cowboy singers) or Keith Carradine.
The cowboy accent does have a drawl -- but at the same time a clearer enunciation, by and large, than most Southern accents. I call it "Texas meets Minnesota." The distinctive cadence is decidedly un-Southern. And one thing you'll hear more clearly from a cowboy than from a Georgian or Tennessean is the R after a vowel. A Southerner will say "yo'." A cowboy, "yore."
Sam Elliot's deep baritone voice and slightly slower speech may seem to the unwary ear to be a more typical Southern drawl, but the cadence is still cowboy, recorded at 45 rpm and played back at 331⁄3. Besides, Sam grew up in California and Oregon.
The truth is, accents evolve continuously, sounding different from one generation to the next. Today's Texas accent and modern cowboyese share an ancestor, but having split off long before the advent of the phonograph, let alone radio and television, they've followed different evolutionary paths.
Cowboys on the northern plains were more likely to meet up with people speaking the ancestor of today's upper Midwestern accent as heard in Fargo, while Texans had the Southernness of their accent subtly reinforced rather than subverted.
Since my mustache's recent growth spurt I've been working on sounding less Southern and more western. For when we move back to where folks don't complain about bad weather, but brag about it.