They were "found safe" when Jackson was pulled over in his car in neighbouring Oklahoma.
If the dad is convicted of the capital murder charges, he faces the death penalty or life in prison without parole.
Thompson said his office is still considering whether to seek the death penalty and would not discuss a possible motive or how the boys were killed.
Police officers went to the family home on Saturday afternoon after one of the siblings didn't show up for a soccer game.
In September, Jackson posted a series of chilling messages on his Facebook page.
"Be nice to the demons, they make decent pets," he wrote on September 13.
On the previous day, he posted: "You can be as deceptive as you want, but you cannot hide the nature of an animal."
And on September 10, he wrote: "When you have to lie to the mother's body to carry your seed, do not expect the abomination to succeed."
Earlier, he shared a meme which read: "BLM: Baby lives matter."
Jackson filed for divorce from his wife, Tara, on August 20, 2019, according to court reports seen by The Sun.
Their next hearing was scheduled for November 12, 2020.
On September 4, 2020, records show his lawyer was granted permission to withdraw and Jackson began representing himself.
Neighbours said the couple lived separately, KCTV5 reports.
They said the two boys lived with Jackson at the rural home while the couple's other children – the two girls and an eight-month-old baby – lived with their mother.
Jackson reportedly visited his wife and other three children often.
