Ok mrcimino1, you were right on the money! (unfortunately)
I tried again tonight (seems the agents working the late shift are the most helpful). Agent was friendly, helpful, and had all three of my standard awards set, names input, etc....then got to the pricing step. After some hmms and aaghs.....he went "uh oh, no awards travel in June for Korean Airlines...). He explained to me why, and basically (as you said) Korean shows availability but blackout date range blocks the final pricing of the ticket.
However, he then spent another 20 minutes on the phone with me, finding me dates on July 1 (instead of June 17) and return on 7/19 (instead of 7/13). So basically instead of getting three free tickets for my relatives and paying for my wife and I, my wife and I will use the awards tickets and I will chip in with the relatives on their 3 tickets. Only thing is, I lose the 14k miles toward my 2006 status since I am using the awards ticket myself (my relatives have a more restrictive travel schedule than my wife and I and want to stay 4 weeks, but my wife and I only want to stay 2-3 weeks).
So all is well that ends well....save myself $2700 by getting two standard awards, even if it skews our timetables a bit.
Thanks for the input!