Hotels should accept credit cards. Capitec is your friend. The airport will have boatloads of USD.
Unlike SA, people zip in and out of Incheon Airport all the time so you should be able to exchange there. So I'd exchange half here, rely on a credit card (not Standard Bank or Absa), and exchange the rest at Incheon Airport.
SA = exchange rate = hellishly expensive wherever you go
SK = Incheon Airport is much better than anywhere in SA. When you're living there: just outside an Itaewon station exit (3, I think?)
You can also post in the South Africans in South Korea fb page in case someone here has won and do a direct exchange (keep it secret) but with Covid I don't know how good your chances will be.
Edited to add: Ask questions, because in Korea it's often what they not say that's important: do you have to go through the hotel because you're not allowed to exchange money at the airport?
Or do you have to go through the hotel because they'll make a lot of money out of this, and save on banking fees, and they're quietly diverting you.