I havent seen this mentioned in Texas but what about San Antonio --
I am not sure about how many "dog" parks there are but I know the Riverwalk is dog friendly - lots of shopping and historical sites.
But hotel wise -- how about starting with a hotel that is dog friendly and checking -- suggestion -
Drury Hotels Official Website - Locations & Reservations
This hotel allows dogs to stay free and very dog friendly. Sam has always been allowed to go every where in the hotel.