The Epic of No-China has taken a tentative step in the positive direction.
After calling the helpful people at Southern China Airlines, they informed me that I can only cancel or exchange my now-useless flight to Beijing through the cheap online ticket agency I bought them through. So I called the customer service number, and after many attempts, I got through to a real person.
Turns out that I can cancel my flight, but the company won't give me a refund. What they will do is give me credit with the airline for a future flight. Actually, this is alright. While I won't be able to travel this vacation (it's Chusook - Korean Thanksgiving = too many people traveling = too expensive), I still have 3 months here to use the ticket. I, of course, need to confirm all of this.
So with that sorted, I can choose another destination for later travel. Japan? Tokyo? Singapore? Kuala Lumpor? Hong Kong? Any ideas?
Where do you think I should go?