You have to define first what you want to do in Texas.
Fort Worth has grown a lot in recent decades. It has more of the cowboy vibe than Dallas but you're still talking about a city of nearly 1 million people. The Fort Worth Stockyards area is the best place for a Western vibe. However, I'd choose Abilene, Amarillo, Lubbock, or San Angelo to get more of a Western feel.
There are some dude ranches near Bandera, TX in the Hill Country State Natural Area. This area is about 60 miles from San Antonio International Airport. Dixie Dude Ranch, Mayan Dude Ranch, and Rancho Cortez are the bigger ones out there. You can do some amazing horseback rides and swim laps in pools there. I know Dixie and Mayan have good lap swim pools. The dude ranches are a good place to escape and unwind. The downside of the dude ranches is that the guests are mainly couples and families so some unattached, alone guys might feel out of place. However, most travel stuff doesn't cater to single people traveling alone so the dude ranch isn't that bad of an option.
You could also go out to West Texas. West Texas has Big Bend National Park, Monahan Sandhills, and the Guadalupe Mountains with Guadalupe Peak (highest point in Texas) and adjacent El Capitan. Guadalupe Peak is about a 6 hour hike, so expect to spend all day with it, especially since the nearest hotels are in Van Horn, Texas (about an hour from the base of the Guadalupe Peak).
The dude ranch thing is probably the most authentically Texas/Western thing that you can do and that's an easy 3-4 day trip. The West Texas stuff, especially Guadalupe Peak, is more fitness oriented and could be more isolated. Midland-Odessa is the nearest airport to that stuff.
Don't come until October. October is the best month of the year to do the dude ranches or West Texas mountain ranges/Big Bend stuff. Anything you do in Texas will be miserably hot from now until October 1, so please wait and you'll have a better time.
