"I would just ask about it."
Translation:Ich würde deswegen einfach mal nachfragen.
You could also say "Ich würde einfach danach fragen.". It's simple and more natural. In this case, "just" is translated with "einfach" which doesn't exactly take the meaning of "simple" in this case. "danach" isn't used in its temporal form ("after"), but more "literal" as "nach der Sache" = "about the thing" = "about it".
kyky, here I am replying to myself, again to beat the right margins. I have nothing against your translations of deswegen. But if you look at the English equivalents given in dict.cc such as: therefore, hence, because, ..., none of those are in "I would just ask about it." or even suggested by that English sentence.