Without sounding too cliche, transfer the UR and book the tickets ASAP. You're not vacationing, you're visiting sick family. You cannot afford to have the award seats disappear. Surely you're under enough stress as it is with your personal situation. And to me, there is no better use of miles/points than when you need them for something urgent. Transfer the UR, book the award now, and be with your family.

And also, just as a note, even if something was available on Cathay Pacific (CX), it would have to go as two awards, as AA only allows routings from North America to the Indian subcontinent that cross the Atlantic.