How poorly did I play this hand?

[kingofmirrors]

9 players remain, final table is made at 8. Blinds are .75/1.50. Average stack is $29, but most people have less because the leader has ~80. My table is 4 handed.

UTG calls
Me (Button, ~$25) raise to $4 with Ah9c
SB folds
BB (~$25) calls
UTG folds

Pot size: $10.25
Flop 6h 7h Th
BB, an extremely tight player, bets $4 and his hand is shaking like crazy when he puts it in. I call hoping to improve but knowing there's a very high chance I don't have the best hand.

Pot size: $18.25
Turn 6d
BB bets $4 again, and now I'm pretty sure I'm beat but I'm getting ridiculous pot odds, not to mention if I hit I can probably take the rest of his stack... unless he flopped a set.

Pot size: $26.25
River As
BB pushes for remaining $13. I contemplate and fold, fairly confident that I'm beat but also because I'm out of the tournament if I call and lose. He shows Qh Qd.

Looking back I thought maybe I should have pushed on the flop. But if he made his hand on the flop (I wouldn't have been surprised about 2pair, set, straight, or flush... though the last two much less likely) I'm a pretty significant dog when he calls. If he has something like AT, AJ, AQ, AK, then it's probably about even money unless he has a heart. I'm really only in good shape if he has a non AA overpair, and even then I'm going to get called every time and lose probably 45%ish.

I think the real problem might have been that I wasn't watching the blinds. I didn't realize that blinds were going up after that hand, so leaving myself with $13 was pretty much suicide on $1/2 blinds. I guess that would justify pushing the flop a bit more, but still BB and myself were two of the bigger stacks, probably #3 and #4, and I didn't think a confrontation there was appropriate when 4-5 other players were shortstacked.

Thoughts?