It seems the definition of おしゃれ is something like "stylish", but it doesn't seem right in this case. What does that mean?
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オシャレ is also spelled お洒落. This breaks down to honorific prefix o- + share, the stem of the verb shareru, "to dress up, to get fancy, to be stylish, to be refined", or even "to be witty" in terms of how one speaks. This could be used as a noun, or indeed as a -na adjective, like in your sample text.
You may have heard the term dajare for "stupid joke": this comes from prefix da- for "stupid, worthless" + this same share for "witticism".
So in the context of your quote, "dressed up" certainly doesn't fit, and as you note, "stylish" also sounds funny. But "refined" or "fancy" would seem to work.