OK - even better news! Talked over trip schedule with my relatives, ends up they can leave mid-July and return mid-August. Korean Air blackout date in June is no longer an issue!

Quickly cancelled my existing 2 tickets within the '24 hour' period, and was able to grab 3 standard awards that fit our schedule. $4k in free tickets, only have to purchase two more (for wife and I) and we are all set and travelling on the same flights!