At first the tiny slivers of cold water come in little more than a sprinkle, but the weeping wound in the clouds widens by the second.
This piece is at its best when you're being evocative like this. This sort of metaphor reads spot on for the genre. To me, there's almost no such thing as too heavy-handed with them in a Western. You have a bunch of this but you're almost never going to go wrong with more.
All he can do is hunker down against the horse’s back
Would consider "his horse's back" here, and elsewhere. You could also consider giving his horse a name. Either one of these will reinforce the punch of him losing his horse at the end of the story. You do refer to the horse as "his horse" at the end when it's gone. Might also consider giving the horse gendered pronouns—it's pretty typical for a story like this.
It takes about fifteen minutes to reach the outer bounds of the Cottondale Ranch, named after its owner and proprietor. George Cottondale had moved down to Texas during Reconstruction on the wings of a generous land purchase,
This phrasing is a little awkward from the first sentence into the second. Maybe consider something like "named after its owner and proprietor, George Cottondale. He had moved down to Texas…"
a Confederate who had caught himself a bullet
Love this. Great classic Western phrasing. Plus, we love when Confederates catch bullets!
Then again, one of the other ranch hands probably forgot to close it when they went out. The possibility of one of them hurrying back in the rain with their tail between their legs forces a guilty chuckle out of him. Then again, he was doing the same thing.
The first and third sentence here both start with "Then again" and it's a little bit jarring. I would consider a way to rework one of them so that it doesn't feel quite so repetitive.
What he does hear, however, is a pained baying from far too close for his comfort
Maybe consider something like "pained baying from nearby, far too close for his comfort." To me, "from far too close" is just a little bit awkward- it's a hitch in my reading here.
His warm brown eyes meet her panicked gaze and linger there for a moment. Cow eyes were all too good at conveying human expressions.
Tell us what human expression they're conveying! I know it says she's panicked, but I think this is a missed opportunity to give us a little more emotion.
he’s already on his feet by the time the dread crystallizes like a knife in his gut.
This reads awkward but I think the imagery is great. I think take it out of simile and into metaphor- "the dread crystallizes into a knife in his gut".
His already pale mass of blobby flesh is paler than it had ever been in life, his clothes gone.
The pale -> paler line feels a little bit repetitive. Consider rewording one.
And now he’s dead, and all Dominic can manage is a choked gasp.
That’s all the man in gray needs to hear. He startles and whips around, but Dominic’s already bringing his gun up and pulling the trigger.
I'd split this after "And now he's dead." Give that line a chance to punch. In a lot of other writing it would be too much, but in this genre it almost feels wrong somehow not to have that be the line on its own. Then make the next line something like, "All Dominic can manage is a choked gasp, which is all the man in gray needs to hear. He startles…"
His eyes are steely and deep-set in his gaunt face, all stretched skin and unpleasant angles. Looking down, his frame is wiry, the muscle of a man whose body is accustomed to eating little. When he speaks, his voice is gravelly and dripping with contempt.
Hell yeah.
His eyes are hazel. Looking into them, Dominic gazes into another world. It’s a world of hatred, built on hatred, coiling in on itself like a modern day ouroboros. It’s a world of conviction rooted so deep into the ground that it goes straight down to Hell. It’s a world of shame, like liquid metal boiling under ice.
FUCK yeah.
Carter was a man with an awful lot of anger
You lapse into past tense here, which is a little jarring, and I don't think you need to reinforce his name here.
So he picks up his revolver.
Awesome.