"Uporządkować" can be "sort out" too, though I'd more often translate it as "tidy up"; "zająć się czymś" is roughly "take care of something", maybe you could use "occupy yourself with something" if it's more in the time-wasting sense.

I stand by "sort out", or even just "sort" as a good translation for załatwić, especially in British English, where I would use "sorted" as a single word sentence similar to "załatwione". There's other possible translations too, I could e.g. imagine using "score" when it's specifically about obtaining something physical, or "get it done" if it's about expediency or perseverance; if anything there's too many ways of translating it!