Since I haul my own horses into Yellowstone, I really can't speak for any of the guest ranches or what they offer. I just refuse to rent horses when I can ride my own. But I understand that hauling your horse from Texas up and back would add several days to your trips.
There are concessionairs that offer horse trail rides in about 5 places within the park, as well as numerous dude ranches around the perimeter of the park.
Any of the bedroom communities will have ranches that offer a multitude of services. Jackson Hole, West Yellowstone, Cody and Gardner are just out side the gates of the park as well as communities that are little farther away such as Ashton Id. Jackson Hole is my favorite. Good airport. Good food and night life. Close access to Grand Tetons and Jackson, Snake river for white water raft trips. Horse rides, hunting, fishing, skiing in the winter. A great town.
Flag Ranch is located north of Jackson Hole between Grand Teton and Yellowstone Parks. I often trailhead off the Road that runs between Flag Ranch and Ashton Idaho. Lots of trails in that area.
Jackson Lake Lodge has horses. It is also the starting point for the Two Oceans Trail up to Grand View. It looks like much more modern Hotel than the historic hotels inside the Park.
There are several guest ranches over near Turpin Meadows along the gravel road that follows the Buffalo river. Turpin Meadows is the trail head to the most remote spot in the lower 48 states. Hawks Rest.
You can google Guest Ranches or Dude Ranches and find a long list.
Here are a couple Jackson Hole Dude / Guest Ranches Jackson Dude Ranches, Wyoming Guest Ranch Vacations - AllTrips
This is a view off the Two Oceans trail near Jackson Lake
Crossing the Bechler River in the SouthWestern part of Yellowstone
Hope you have a great trip. You will make great memories