Police eventually shot dead the attacker outside a cinema after his rampage in the cities of Midland and Odessa.
Michael Gerke, the Odessa city police chief, said three of his officers were among the injured.
Road traffic officers initially tried to pull over a car on the Interstate 20 highway after spotting a fault on the vehicle.
But as they approached the driver, he pointed a rifle towards the rear window and fired several shots at the patrol unit, injuring one officer before speeding off.
As he drove he fired at passing vehicles.
David Turner, the mayor of Odessa, told Fox News: "A friend of mine passed away. This coward pulls up beside and opens fire on the car where the man and his family were waiting at a stop light."
The man switched vehicles to a postal van and continued his rampage before being halted in Odessa.
The attacks come less than a month after a gunman killed 22 people in the Texas city of El Paso.
The latest bloodshed immediately ignited fresh calls for gun control to reduce the prospect of mass shootings.
President Donald Trump tweeted that he had been briefed on the incident by Bill Barr, the Attorney General.
"We need to end this epidemic," former Texas congressman Beto O'Rourke reacted on Twitter after the latest killings.
Another Democratic contender, the former San Antonio mayor Julian Castro, appealed on Twitter to the Republican-controlled Senate. "What is the number? How many Americans are you willing to sacrifice to the National Rifle Association?"