I have three ideas about this(all about 2nd of April):

1. You buy one of the two morning flights on flyasiana.com/english that show availability. This works if you get around signing up for their mileage programme and have a creditcard that works with that crazy website(I havent), Might be a relative safe bet to fly on Asiana. But it seems to me as you have tried that already.

2. Do you know someone who speaks german? I always book my korean domestics on OZ through the german expedia.de. The website shows availability for the afternoon that day on the 2:55pm and the 8:05pm flight after some trying - I could book these flights straight away, not sure though if it will go through. The tickets are issued automatically right on the spot and money is taken by Asiana in Frankfurt. Price there is 70$ one way.

3. Do you know someone who speaks korean? Then book the flight on Jeju Air, they have availibility on their 1:35pm flight at 50% off, so the price would be just 40$ which makes it excellent value and they are a great airline.

Hope that helps in some way. Flights to Jeju are seriously packed during the first weeks of April and in general on Fridays and Sundays, so it might be true that the seats are somewhat taken(if only by travel agencies making good money on that). Jeju is a great place (Ulleung-do is even better though)!