The translation is not very good. It means that you will take care of something, meaning you will solve it. Other examples:
O să rezolve ea cu biletele. - She will take care of the tickets (she will buy them or somehow get hold of them)
Rezolvi tu cu mâncarea, te rog? - Can you please take care of the food? (can you make sure that there is enough food etc.)
You're being nice, saying that "the translation is not very good": that is not the wording that first came to my mind! :-) Thank you for your explanation anyway! Here's another suggestion for a translation: "I will deal with the clothes". Do you think that "rezolv" goes further than just "deal with" or "take care" does?
Yes, that's fine, I would say. „O să rezolv eu cu hainele” implies that something needs to be done with the clothes (for instance, that they need to washed) and that you will be the one who will get it done. This means that there is a problem or an issue involving the clothes and that you will solve it. This is pretty colloquial, though.
A good translation for "I will find a solution for the clothes", besides the one already discussed, could be: „O să mă ocup eu de haine.”. It basically means the same thing and is also colloquial, but widely used.
This is meaningless nonsense! Maybe it's supposed to mean "I'll look after the clothes" or "I'll deal with the clothes." "I'll take care of the clothes?" There are plenty of options that are comprehensible English ..... why choose one that isn't???