Are the sun and moon generally personified in Korean folklore, or just in this origin story?

No, not as far as I know of.

Most Korean folk tales are similar to European fairy tales in the sense that each story may share similar themes and motifs, the specifics don't seem to relate. I could be wrong, but this seems to be an independent folk tale rather than part of a continuity of a sort. The sun and the moon don't normally make an appearance in most folk tales I remember. However, there is one notoriously famous Korean folktale I forgot to link called "Bulgae" (불개, lit. "fire dog"). It's part of a larger Korean story about the solar eclipe. Here's the general gist of the "bulgae" section:

Once upon a time there's a kingdom of darkness that wants light. The king sends a dog to catch the sun and bring it, but when the dog catches it, the dog's mouth burns and lets go of the sun. The king was undeterred, and sent another dog to catch the moon. The dog caught the moon, but had to let go of it, because it was freezing cold. The king keeps on trying though, and that's why we have solar eclipse.

The important note you'll see though, it that the sun and the moon are NOT personified in this story, the sun is literally just a blazing fireball, and the moon is a freezing ball, rather than being personified.

I have a second example, but I couldn't find another story posted on it, so I'll ask it as a separate question.

But yeah, from my knowledge, the sun and moon aren't typically personified, since people don't really "pray" to the sun and the moon. There are also a lot of conflicting stories regarding each the moon in particular, like how there are bunnies on top of the moon making tteok (떡, rice cakes) in certain Korean folklore.