the JAL website is best for both JAL and CX (Cathay) availability - you can just search via there and then call it into AA when you flight the flights you would like. you'll probably need to book to BKK for best award availability and then book separate one-off flights via cash or whatnot for the short flights to HAN/HKT.

for the shorthaul flights, if you have any starwood points, there is a promotion for booking bangkok airways flights using JAL miles for very cheap for flights through 12/31/14. (you would transfer starwood into JAL assuming they have availability and then call JAL to make the booking)