Of the three translations you mention, I would not use resultar. As @CesarGon mentions, acabar and terminar are common words to express what you want.
¿Cómo acabaste mudándote a Kansas...? / ¿Cómo terminaste mudándote a Kansas...?
¿Qué terminaste haciendo para la cena anoche?
Simplemente terminé trabajando aquí, no estoy seguro cómo sucedió.
I would say both are correct and people will understand you either way. However, it seems to me, as a native speaker of Spanish, that terminar is more common and/or more natural, although, as usual, there might be regional differences I'm not aware of. I cannot really explain why I think it's natural with a strong argument, it's just something intuitive: "it feels better" using terminar. It might be that terminar has a stronger connotation of an ending (of a final state, as you said), while acabar has other meanings as well. I mention three, for completeness:
- "It just happened" can be translated as "acaba de ocurrir", so acabar can be used in the context of something recent, but terminar cannot be used like that. In your example of "winding up having this job", that's a question that you can explain years after you actually got it, and in that case you would say "terminé trabajando aquí"; if you got the job recently, you can also say "acabé terminando aquí".
- Acabar means to have an orgasm, but terminar does not.
- Acabar means give the finishing touches (for example, to a painting or a new building), in the sense of putting attention to details, try for excellence. In contrast, terminar una pintura has a connotation of finish it already, that is, do it soon and don't mind the details/quality. When an object (especially a hand-crafted object) has been perfected, you say that the object "tiene un buen acabado".
So, to summarize: both are correct, but go for terminar if in doubt. It is closer to your intended meaning.