Last January, we went to Big Bend for a week and the weather was beautiful--sunny every day and highs in the mid-70s. Perfect weather for camping and hiking. We also visited Ft. Davis NM and the McDonald Observatory, where it was a little windy, but otherwise nice.
Several winters ago, we camped in Texas for a month during late January and early February. The Hill Country around Kerrville, Fredericksburg, Junction, and Uvalde, where we stayed at South Llano SP, Garner SP, and the city campground in Kerrville, was cool, but sunny. We loved the bird watching at South Llano, kayaking at Garner, and visiting the Nimitz Museum of the Pacific in Fredericksburg.
We also stayed on the coast at the North Padre Island NP cg and Goose Island SP near Rockport. Weather at NPI was cool, cloudy, and very windy, but the weather turned really nice when we visited Aransas National Wildlife Refuge while the whooping cranes were there. Lot to do in that area beside wildlife viewing, including the aircraft carrier Lexington in Corpus Christi, boat tours to Matagorda Island, fishing in the lagoons, buying fresh shrimp and fish right off the dock in Rockport, etc.
We have lived in Texas in the past and do not like the summers anywhere--too humid. But we find the late fall and winter to generally be nice in the southern and far western parts (except during hurricanes!) and spring in the Hill Country (except during long rainy periods) to be very nice. For the rest of the year and the rest of the state I much prefer Colorado. Guess that's why I have lived here for 54 years.