Adams, who never denied killing his son in February 2002, offered no final statement before his death by lethal injection Tuesday, the Houston Chronicle reported.
Adams, 42, shot his son, Timothy Adams Jr., during a 2 1/2-hour standoff with police at his home. His attorneys maintained the shooting occurred during an emotional crisis brought on by news that his wife intended to leave him.
His attorneys said their client had no prior criminal record and was an exemplary prisoner during his seven years on death row.
In the witness room, Adams' mother, Wilma Adams, repeated, "He's going to sleep. He's going to sleep. He's going to a better place. He's going to get to see Jesus," the Chronicle said.
As Adams' execution date neared, members of his family and church appealed to the Texas Board of Pardons and Paroles, the newspaper said. The board rejected the inmate's appeal Friday.
Adams' appeal to the U.S. Supreme Court mirrored a failed appeal to the Texas Court of Criminal Appeals. The federal high court rejected his appeal several hours after it received it Tuesday.