Just to follow up on my story-
I have been calling AA everyday and using Expert Flyer to check non-stop TLV-NYC availability. Today I see tons of flights for the dates in May I was looking at- just like that. And of course, I reserved them. Now, I'm just hoping my Starpoints or CC points post in time before the reserve is up.
By the way, when EF says 4 seats available, then I think AA gets 2 out of those 4, because for dates that said 2 available AA told me they didn't have. But anyways, as I've said before there is not an exact correlation between EF (ie- Matmid's) inventory and AA's award inventory (both for the bad and for the good- sometimes AA has inventory that Matmid doesn't have, or at least EF says they don't have).
Thanks Dan, and everyone else for their help. Maybe I'll stop in Europe next time (next time that I don't need many bags- which is not so often).