We used metal panels, 3 sided, and i get out there and caulk any holes. I live near Denton and my goats survived the frigid February we had- minus 6 at our house. Ours is too tall for them to jump on, since they would then escape. We have several giant wooden spools and an old holey rubbermaid horse trough for them to climb on. They sleep outside unless it's raining or very cold. My sheep insist on being in the shed at night tho.Between you, me and the proverbial fencepost, those tarps are not going to last long in the Texas sun. Now, your chicken run is a good place to start. It looks solid and too tall for the boys to jump on the top, so that is a good choice. One of the members, how is in TN puts some type of paint on his tarps, but I think he starts out with canvas tarps. @Mike CHS is the fellow. He does sheep.
Yeah, the boys are not going to like getting wet and in Texas you do have the problem of keeping the air flow going and keeping the weather out. We enclose 3 sides, and the roof and position the shelter so the back faces most of our incoming weather (did that make sense?). We started out building cabanas out of plywood with a metal roof. Ours are short and the goats get on top during nice weather. I don't think I'm suggesting using plywood to cover the top and sides as in today's prices that would be rather expensive.
As an alternative you might want to look into plastic roofing... This is what we use on some goat tunnels (gunnels) we built.
It holds up well to the weather, can be translucent, reflects UV, and is not nearly as heavy as plywood to work with. You would want to be creative in attaching it to your wire frame. The goats rub on it and the corners tend to eventually crack off, but that doesn't seem to impair overall function. We also used these panels to put a roof on the dog kennel.The Suntuf 8 ft. Polycarbonate Corrugated Roof Panel is designed for comfort. This durable panel is solar gray, an attractive color that permits light to shine through. The panel helps to protect openwww.homedepot.com
Just some thoughts. You don't need to do all sides, just three sides and the top.