Welcome to the site and sorry about the late reply. I was out climbing...
I imagine by now you have left for your trip but here's some info that might be useful for your trip to Korea.
The rainy season begins in late June and lasts for about a month or so. Sometimes the rainy season is really wet, sometimes you don't know if its really the rainy season. It also depends on where you are in Korea. Though the country is small, the weather varies greatly from city to city.
If it's sport climbing on sandy beaches you want, maybe you should consider climbing in Krabi, Thailand. That being said, try Busan. There are some climbs in Annam Park in Busan where you are right next to the sea. Also, Busan has a lot of climbing to keep you busy. There's a lot of cultural things to do there as well. I haven't been to any swimming beach in Korea, but i know there are some in Busan. They might not be what you'd expect. I know they get quite busy in Mid-summer.
There is also a lot of climbing in Ulsan which is near Busan and the sea.
My advice, given your criteria is to head southeast and start in Busan. There are lots of climbers in that area and a few gyms where you can meet folks. You can also use this forum to plan trips.
To retreat in dynamic fashion from a climb.