Traveling to Korea During Chuseok?

I booked a trip to Korea for our family on 9/30/25, not thinking about the Chuseok, which falls into our traveling period.

Supposedly, the next Chuseok falls into the Golden Holidays (황금연휴) too...

Here are my main concerns/questions.

1. Will there be limited availability for KTX tickets traveling from/to Seoul?
2. How about restaurants? I assume lots of places would be closed during this period.
3. How about a skincare clinic?

I'm debating whether I should change my flight ticket to a later date but I have already changed the dates once for a $150 change fee.