The Governor will promptly be ousted. While the German settler community in the state was Unionist, the overwhelming Majority of white Texans were not, in fact it was perhaps the most Pro-Confederate state by some metrics, potentially only beaten by South Carolina by some accounts.
For reference 75% of the state voted for Breckinridge to Bell's 25% in 1860, add to that the basic fact that Bell's "Unionists" were more often than not no less secessionist than their Constitutional Democratic counterparts and we're talking about less than one in four white Texans being Unionists, probably even less than that.
Sam Houston won on account of the fact that his Unionism was pretty much meaningless, no one with more backbone than "This isn't going to end well" could possibly be elected, and under no circumstances would have support when the militia launched its coup.