I agree with you. I do not see any evidence in scripture which would indicate that Saul killed anyone. He used his position to seek proper authority (within the Jewish hierarchy) and we see him with letters on the way to Damascus, that is, with some kind of written authority to go there.
It is something I have tried to research previously. Was he using his liberty as a Roman citizen, which he had because he was a native of Tarshish ? Did his Roman citizenship give him some kind of ability to go outside of Israel and into Syria in his quest to persecute Christians ?
I can't say. I have not been able to research this far enough to find out.
But what is true of Saul, I think it is important to say, is that he 'verily thought that he ought' to be doing these things. None of what he did was unlawful, in Jewish terms or in Roman terms, the two authorities over him, naturally. And all of what he did he says he did 'in good conscience'.
He thought it was right to do it.
Only the experience on the road to Damascus could ever have stopped such a zealous, energetic, 'Pharisee of the Pharisees' in his course of eradicating Christianity from the face of the earth.
His real state of soul is seen in Romans 7, however. Sin had been discovered within his own self. 'Thou shalt no covet' had revealed it within him.
'O wretched man' he says of himself, and admits that that was the condition inside himself, even as he persecuted others, until the voice of Jesus Christ himself spoke to him - 'I am Jesus whom thou art persecuting'.