Among the victims is a 17-month-old girl, apparently shot in the face, said Eric Finley, marketing director of Texas' UMC Health System. He said she was in satisfactory condition at a hospital in Lubbock.
The Texas Department of Public Safety said in a statement that the state trooper and two officers were stabilized.
President Donald Trump was briefed on the attacks and had been monitoring the situation, White House communications officials said.
Gov. Greg Abbott called it a "senseless and cowardly attack."
The attack, the second mass shooting in Texas this month, prompted a round of calls for stricter gun laws from some hopefuls running for the Democratic nomination for president.
U.S. Sen. Kamala Harris, D-Calif., said on Twitter, "America is sick of this. We need to act."
The new laws will allow licensed handguns in houses of worship, and they'll let gun owners transport and store their weapons in parked vehicles on school grounds.
After initial reports of an active shooting in Odessa, police urged the public to get off the road "and use extreme caution."
Midland police initially said on social media that they believed there were two shooters in two different vehicles. "The two vehicles in question are: gold/white small Toyota truck and a USPS Postal Van."
Dennis Romero writes for NBC News and is based in Los Angeles.
Andrew Blankstein is an investigative reporter for NBC News. He covers the Western United States, specializing in crime, courts and homeland security.
Tom Winter is a New York-based correspondent covering crime, courts, terrorism and financial fraud on the East Coast for the NBC News Investigative Unit.
Michelle Acevedo, Suzanne Ciechalski and David K. Li contributed.