I'm not sure what you mean by this. No TX legislature is 'under the thumb' of a TX guv as they are largely ceremonial with no actual power. The Lt. Guv is the one that runs the lege and thereby wields the power. In Short Count's case, he was working with Bob Bullock, who although putatively a Democrat, isn't what you would recognize as a Demo in DC or NYC. The Democrat party in Texas is basically dead except in name and in a parts of a few urban areas.
You could probably use a rule of thumb in identifying political species in TX: "if you're white and a Democrat -- you're really a Republican." The only liberal Demos I can think of are all Black.
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Joseph Noonan Houston, TX jfn1 at msc.com
haiku for struggle
democracy rocks noisy protests in the streets frowns high in rich suites
-Chip Berlet