Monica Bracknell, an 18-year-old senior who survived the shooting, stopped to tell the governor that the attack should not be used as a political push for gun control.
He spared others saying he wanted his story told.
Police said Pagourtzis confessed to Friday's killings after he was taken into custody, but authorities have offered no motive yet for the massacre, the fourth-deadliest mass shooting at a US public school in modern history.
"From a Republican perspective, you're not going to lose the Republican primary by being too pro-Second Amendment", Jones said. The suspect was taken into custody.
Two Obama administration figures, former Education Secretary Arne Duncan and assistant secretary Peter Cunningham, called for a nationwide school boycott to push for tougher gun-control legislation, while others cautioned against blaming firearms. Instead, they are looking for leaders to help the country through a slow-rolling epidemic, one that has claimed almost 175,000 lives since Sandy Hook, and so many other kinds of victims, as well.
Abbott is running for re-election in November.
"People are making this into a political issue", she said.
If I, as a parent of teens, allow their friends to drink at my home, negligently let them leave in their vehicle and they end up hitting and killing others, I am legally responsible for the activity.
Trochesset, the sheriff, said his granddaughter was inside the school, three doors down from the gunman. "That's what you call a hero".
He added: "And her best friend that spent the night at my house, swam in my pool, is dead from a tragedy".
Authorities have identified the gunman as Dimitrios Pagourtzis, 17. The chaotic day devastated this small, conservative city, where everyone knows their neighbors and just about everyone owns a gun.
Students are checked before entering Santa Fe High School David J Phillip PA
Texas has been among the states most strident in its support for gun rights, even as mass shootings have caused other states to slowly start tightening laws.
One Santa Fe school police officer who responded to the attack was shot and remained in critical condition Sunday, according to the University of Texas Medical Branch.
The 17-year-old suspect has been jailed on capital murder charges. He has not entered a plea.
Ms Rodriguez said Mr Pagourtzis had been increasingly aggressive until her daughter stood up to him, embarrassing him in class.
Baze said she believes the incident embarrassed the suspect so much that he targeted Shana.
She says she still hears the bullets and remembers the "fear on people's faces".
The student's aunt, Candi Thurman, wrote on Twitter: "Shana turned 16 on May 9th".
Another student killed, Jared Black, turned 17 last week and was supposed to have a birthday party Saturday. Teachers Glenda Ann Perkins and Cynthia Tisdale were also killed.
Schools in Santa Fe were closed through Tuesday and a crisis hotline was set up for traumatized students and parents.
"He's had ups and downs". They appreciate the thoughts and prayers, which help them through the darkest moments.
Officials said that teachers and staff are scheduled to return to campuses on Wednesday, while students are scheduled to return to campus on May 29, according to KPRC-TV.