In the last seconds of the video, Blair - holding a knife but already struck several times by a stun gun - appears to stumble toward the deputy standing in the bathroom doorway at the family's Rosenberg-area home.
The deputy shouts for the man to stop, then begins firing.
The video appears to show the deputy continuing to shoot almost a dozen times, the pistol following Blair as he collapses to the floor.
Anguished screams from family members can be heard as they are hustled out of the home.
Community activist Quanell X on Wednesday called the shooting "nothing less than a cold-blooded execution."
Blair's family gave the video to the Houston media and said they intended to hand over a copy to the U.S. Attorney's Office.
It was never provided to local investigators, so was not taken into account by a Fort Bend County grand jury that declined to issue an indictment in the case.
A Fort Bend County Sheriff's official said the department could not comment on something it hasn't seen.
"We consider it evidence and it needs to be turned over to the proper authorities," said sheriff's spokesman Bob Haenel.
The deputy, an 11-year veteran, was sent to the home in the 7000 block of Dawn Bloom on Nov. 4, because Blair had locked himself in a bathroom and was threatening to harm himself, officials said.
Blair's family said they called 911 only after several hospitals told them that was the proper course in the case of someone who was possibly suicidal.
"All this family wanted was somebody to help. This young man did not have to die," the activist said, standing outside the U.S. Attorney's Office in downtown Houston.
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