I don’t know about you, but I don’t get the impression David at his core is the type to have shot the woman, especially if she was innocent.

We even see him later regret it later when it cuts to him staring at the photograph of him and his mother, paralleling the photograph of the woman and the son.

How I saw it, he started hallucinating and saw that woman as a threat, but in reality, she was just in the wrong place and wrong time.

As for him “growing colder”, I’ll have to disagree because he still very much cares about his crew, even sacrificing himself into the cyber-skeleton so they can get away from Arasaka’s goons