That’s at the end of rainy season and could be very wet, or it could be hot and sunny. Hard to tell so planning an itinerary for this time is difficult. I would recommend being flexible in your travel plans so you can enjoy the sunny days and stay inside during rainy days.

This article has a few one-week itineraries in it. https://joelstraveltips.com/one-week-itinerary-for-korea/

I would recommend visiting Jeju Island even if it is raining. It’s a beautiful place and you can enjoy so many summer activities there, such as water sports, trekking, horse riding, scuba diving, and sitting by the beach.