Let's take another view of the situation (i.e. cheat):
Some unlikely confluence of events (Al Smith wins in '28, some progressive Republican wins in '32 and pushes through a New Deal?) means that the Democratic Party stays powerful in Texas and reactionary at large. The Republicans don't take off in Texas as quickly as OTL - big oil remains Democrat-friendly in Texas, even as more Republican information-tech and banking etcetera businessfolk begin to proliferate. By 2002, though, a red sun is on the horizon.
That year, Austin's Carole Keeton Strayhorn and Pampa's Warren Chisum face off for an open Senate seat. One's a liberal Republican with a focus on fiscal prudence and a penchant for showwomanship. The other is a rural conservative Democrat with a reputation for hardline moralism. After a protracted recount - who knew there were so many Democrats in Webb County? And how did Henry Cuellar get Chisum's old job as Land Commissioner? The world may never know - Chisum scrapes ahead.
The next six years see an oil glut, a budget crisis, and a showdown with the Feds over gerrymandering - in short, the perfect storm for the Republicans. Maybe the Governorship was out of reach, but Warren Chisum had been an uncompromising and ineffective ideologue in the Senate. Who better to win the nomination than a longtime activist and a Strayhorn ally? Someone who promised to put the hammer to corporations that screwed hardworking Texans and the political establishment that aided and abetted them? Someone palatable to everyone from Michael Dell to Royce West?
It had to be David K. Cobb.