Leaving people in the dust if they've outlived or outsucked their usefulness isn't really lawful or chaotic in itself. 'Lawful' indicates a presence of strong principles. If somebody is lawful, that would indicate that they do care about something rather than that they don't. Lawful people tend to be method-oriented, the how of something matters deeply to them. Chaotic people are more interested in the end result, and how they get there is really something that's not terribly relevant.
Also, I somewhat disagree with the idea of an alignment shift in general. An alignment should represent the underlying driving force of your character (greed, pride, bloodlust would be good examples of underlying motivations that result in evil, for example). An integral part of your personality. There is a fundamental difference between those who just genuinely don't give a fuck who has to die to further their ambitions and, well, regular people who just figure out what seem like good reasons for why they do shitty things. If you're motivated to be a dick to people because you don't trust them, that seems like a strictly neutral antiheroic motivation rather than what you would expect from somebody properly evil. Deep down all evil people harbor a monstrous core that sets them apart from others, you know?