“If there’s another shutdown, it will be a cataclysm that will go way past the normal left/right divisions,” Mr. Roberts said. “I can tell you from being in D.C., all eyes are on Texas right now.”

In recent weeks, Mr. Abbott has vacillated between denying another state shutdown is imminent and warning Texans that it will be necessary if things don’t improve…

Some local Democratic officials, meanwhile, are now beginning to test the limits of Mr. Abbott’s authority. In Hidalgo County, on the U.S.-Mexico border in the Rio Grande Valley, which has seen a death rate this month six times that of the state, County Judge Richard Cortez, a Democrat, said Monday he will issue a new stay-home order for nonessential employees.

Mr. Cortez acknowledged in an interview on local radio station KURV that such an order contradicts the state’s reopening—so he has no way to enforce it. Asked whether he expects a punitive response from Austin, Mr. Cortez replied “I don’t know.”