How do I leave Korea temporarily but have ongoing legal and financial matters that require someone to act on my behalf?

You can set up a power of attorney (위임장) in Korea to let someone handle legal or financial matters while you're away, and yes, you can grant it to a Korean citizen — it doesn't have to be another foreigner. For most banking or administrative tasks you prepare a written 위임장 specifying exactly what powers you're granting, attached to a copy of your ID, and many institutions want it notarized at a notary office (공증사무소) to be accepted. If you'll be abroad when signing, you can have it notarized or consularized at a Korean embassy; because banks and offices each have their own format requirements, ask the specific institution what they need, and the Korea Legal Aid Corporation offers free guidance.