final table hand - how was my play?

[ACynicalOptimist]

we just made the final table.
600-1200 blinds, with I think a 500 or so ante.
My stack is 13k, which is low enough where almost everyone at the table has me covered but not by much. There's only a couple big stacks, including villain in this hand who has about 24k.
The hand previous to this, Villain (around 35k) open raises early position, and bb (around 10k) pushes, Villain calls and shows Q-6 offsuit. BB shows 10-10, and so BB doubles up and now it's the very next hand. Villain (with now about 24k) opens the pot with a 4k raise. I'm 3 off the button with A-T os. The blinds are moving up to 800/1600 in 2 minutes. I pushed all-in.

So, given all of the above, was my push correct? Here are some other things to consider: the other players at the table were not very good and so avoiding gambles is good if possible . However, does A-T have enough of an edge over villain's range that this has to be a push? What do we think our fold equity is (0?) given the previous hand and the current stack sizes.

This hand is really bothering me. Obviously since I'm posting about it, villain called and busted me, all I'll say is that I wasn't a huge favorite or a huge dog. It was basically a race. Even though villain could've had a much worse hand than it turned out, I guess I should've known villain could have and would've called with something like a small pair, in which case my stack size may not have been small enough where I needed to gamble my tournament life, especially at the final table where I could've hung around and made some more money while waiting for a similar/better spot.

Argh, anyway, commentary appreciated.

[ssnyc]

with blinds that high and with your read/example I can't ever imagine not pushing here

[Dunkman]

Same, and also, with the overlay from the blinds and your stack size, you should have been ok with taking a flip at this point since you've run out of room to maneuver.