Looks like we're in a similar area. I have a prebuilt shed on property that's 4x9, and 12 chicks. So, I'm also violating the 4sq ft rule. I'm working with a college student budget, so I've consigned myself to working with what I have and having the run accessible 24/7.
I'm in the grasslands too, so I know what you're dealing with as far as wind. This is all hypothetical and hasn't been stress tested yet, but hey, it's something.
My run and shed is somewhat sheltered by a barn, but if you haven't got another structure handy, you might be able to provide a bit of windbreak on bad days with a tarp over 3 sides of the run. That's my plan, anyway. One of those 6 mil clear tarps that only gets put on during heavy winds, so they can still go out in relative peace. In order to deal with the fact that this is essentially a giant sail, I have
18" rebar spikes hammered in at each of the posts, so I can make sure the run doesn't fly off. The shed itself has earth auger anchors, like how they use for mobile homes, and it hasn't moved.
Over the run, I've got shade cloth. It's slightly permeable so the wind doesn't snag at it so bad as full tarps, and it works pretty well to keep the heat off. I have another one over my back patio, so I can actually sit outside during the summer without being burnt to a crisp. Over the top and down the west side a bit helps the best here.
A longer term solution is windbreak trees. I'm not sure if the forestry service is still selling them, so you may have to wait for december, but NM forestry sells them for less than a dollar each, and you can pick them up at your ag extension.
For the coop itself, I've seen a lot of people in hotter climates suggest open air coops. I've seen a lot with just 3 sides and a front made entirely of hardware cloth. Honestly, any search of "open air coop" on here gets interesting threads, but I most liked
this one, and especially
this post of a 3 sided coop that opens up into a run.
The Woods open air coop is a design I see mentioned a lot, too, and I've seen a couple designs here and there.
Personally, I'm trying to decide how many holes I can cut into my shed before it stops being structurally sound, haha.
Good luck!
Edit:
Oh yeah, and I'm feeding / watering in the run, not the coop. Not just to maximize sq ft, but also to avoid mess. My shed's a walk in, and I think I'd prefer that, anyway.