Why did the Engineer not just take another ship immediately?

In Prometheus, the Engineer makes a detour to kill Shaw, whom he could not have known for sure was alive -- she could have been in the Prometheus and before that, he was not interested in chasing after her but rather in flying to Earth. That cost him his life at the hands of the octopus creature who very ironically was the savior therefore of our planet.

EDIT: Some more context: The first time Shaw runs into the revived Engineer whom Weyland wants very much to talk to, the Engineer (I think the scene in the theatrical release and the one currently available on HBO which are slightly different from each other is cut very severely -- it seems like the Engineer acts almost instantly but that may be different from the script) attacks and kills everyone with no provocation can we can tell -- he doesn't exactly kill David but he does pull his head off.

The one exception is Shaw who had noped out of there and was probably farthest from the giant alien -- not sure if we see him consider going after her and then just decided he has bigger fish to fry and starts to prepare to take off and destroy all life on Earth.

Shaw manages to just barely clear the ship as it extricates itself from the soil of the moon where it had been buried for 2000 years and tell Janek, the star ship captain that they can't allow the alien ship to reach Earth -- no point in going home because the Engineer will presumably get there first. The captain very quickly accepts Shaw's plea that he destroy the ship and, having no weapons systems, all he can do is ram the vessel with the Prometheus and this does indeed knock it out of the sky, completely destroying the Prometheus in what appears to be a nuclear explosion which would also obliterate the alien ship but it does not. The alien ship is massive and merely crashes without even breaking up.

Importantly, the captain jettisons Vickers' lifeboat so she can take an escape pod and survive for a couple of years which she really wants -- she strongly and I would say stupidly (or at least very selfishly) objected to destroying the alien vessel and wanted to simply go home. The two copilots die valiantly with their captain but if you figure how long it takes to get to the moon from Earth, she is going to die before being rescued anyway -- with the 3 of them using up the 2 years of air/food much faster. Vickers could have figured that they warn Earth and take the her star ship to some safe place which actually was a reasonable alternative as I imply sort of in my question.

Anyway, Vickers is simply crushed by the falling alien ship and Shaw just barely rolls out of the way of it and makes it to the lifeboat module. She is then contacted by David who is by the way still aboard the alien ship, his head separated from his body -- the Engineer did not bother to further destroy or dispose of him. And David manages to radio Shaw that the Engineer is coming for her.

Shaw is confused: I supposed she assumed that the Engineer died when the Prometheus crashed into his ship but soon she sees what David means: the giant Engineer has pried the lifeboat's doors open and angrily tries to kill her, only being stopped by Shaw's "baby" (and one wonders if the alien octopus creature had any understanding of her as its mother and so sought to aid her -- probably not) which implants an embryo in the giant after a struggle.

So back to my question:

1. How could the Engineer know that anyone was still alive? It occurs to me just as I am writing this that as his ship was crashing he could have been watching what was going on outside or was informed of this by some monitoring system.

2. But even if he knew that Shaw was still alive, which he certainly knew at the time he was killing everyone in the control room of his ship and yet let her escape, why did he suddenly decide that a priority was killing her now? We of course only find out later that there are more ships -- David bargains for Shaw's help with this bit of information -- but presumably the Engineer knew that and he could now confidently resume his plans with no human ship left to stop him. (I don't know why there are other ships and if so whether other Engineers remain in stasis aboard them but there is at least one which they then use.)

3. The only things I can think of are these: Firstly, the Engineer may have been worried that somehow Shaw could interfere with his plans -- maybe she could simply call Earth. Secondly, he may have just been really angry but if so, he would have known that he could leave her to a slow death with no chance of rescue as he wipes out Earth. I think it must be the first -- somehow humans had managed almost to kill him and he was going to make sure that did not happen again.

4. But finally, why did he leave David alive? Disintegrating the android whom he would have known was still conscious would seem like a very good idea if he was worried about tying up loose ends.