Kim,
Check out this link:
http://www.davismountains.com/
You know when you say "ranch" in West Texas you really are talking property. Five thousand acres is considered a small front yard.
I'm pretty sure they have deer and sheep and all the rest of those critters in and around those Davis mountains. The desert does not want for critters. When I went with my father it was mostly to hunt birds, quail and dove. I'm sure I couldn't shoot one of those furry critters.
I think you will enjoy your visit. I think it could be a very interesting place if you go at it with the right frame of mind. Especially if you have a fixed time frame. And thanks for the offer, I'll log that.