Humid isn't the word to describe South Korea in the summer. It's the one place we lived that my father bought air conditioners, because our shoes were growing mold in the closets.

So think hot and very humid, but you will probably miss the rainy season.

Where will you be? The larger cities are pretty cosmopolitan, the rural areas less so.

I'd steer clear of tank tops and instead get a few short-sleeved tops. Shorts and short skirts are fine.

If you are going to be in a big city, most women dress up a bit more than Americans on a day to day basis. You probably won't see sweats or pajama pants on the streets.

However, if you look like a tourist, i.e. you are not of Asian descent and it is clear you are a visitor, most people will be very forgiving even if you do make a slight fashion faux pas.

How exciting! Have you checked TripAdvisor.com ?

I've found vivienne files most helpful for suggestions for a capsule travel wardrobe....not specific for korea, but helpful for how to get a lot of outfits out of a few pieces.

Here is a link that might be useful: Vivienne files

I have also posted on TripAdvisor. That's where I read that uncovered shoulders are not common. Of course my summer wardrobe is all about tank tops and sleeveless dresses. :-(
I will be spending the first two days in Seoul, then taking the train to Busan, which is on the coast in the south and the second largest city in South Korea. My son lives in a suburb there.
I am actually going to take a full sized suitcase instead of my normal carry on size. I have been able to pack 10 days worth of summer clothes in my carry on, so this larger suitcase feels like a steamer trunk to me.