Why it's a mistake: Aside from feeling disgusting, staying in sweaty clothes can lead to a yeast infection or body acne. And although going outside in wet clothes won't make you sick, when you walk out into the freezing air, your wet clothes are going to make you very chilly, and being cold for too long can weaken the immune system.
What you should do: Always pack a clean, dry change of clothes, including undies, a bra, and socks. Shower if you can, but if time doesn't allow for it, just change into your dry outfit.
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