I know that I can use で here to mean by/with but wouldn't it be more proper to use を since よごす is a transitive verb?
しんぐもつちぼこりでよごしてしまいました。
Any help would be appreciated. Thank you!
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Sign up to join this community汚す{よごす} is indeed a transitive verb, but 土埃{つちぼこり} is not the direct object of action here. When you use 汚す, you can specify a direct object(marked with を) and you can also specify the thing that dirtied that object. You can specify that with で. That means, that 寝具{しんぐ} is the thing that got dirty, and it was dirtied by 土埃. Another example:
手を血で汚す - dirty hands with blood