Fold an overpair to a single bet?

[maurile]

$3-$6 at Viejas (B&M). I'm in middle position with QQ.

UTG raises. I 3-bet. BB calls. UTG caps, I call, BB calls. (UTG is a solid player. BB is a fish.)

Flop is T 7 4 rainbow.

BB checks, UTG bets, I raise, BB calls, UTG 3-bets, I call, BB calls.

At this point, I'm fairly sure AA or KK (or both) are out there, and I'm behind. I can't fold to one small bet here, though. I have at least two outs (with implied odds if a Q comes on the turn), and besides, it's still possible UTG has JJ and I'm ahead. It's hard for me to put BB on a hand since he could play two overcards [AK, KQ, etc] this way.)

Turn is a 3, completing rainbow.

BB checks, UTG bets, I call, BB calls.

Was this a bad call? It's hard for me not to put UTG on AA or KK, or maybe TT. I guess it's still possible that he has JJ, though.

River is a J. UTG bets.

Oops. I can no longer justify a call by hoping he's got JJ. Nobody's got 89, so I'm not worried about a straight. I can't put anybody on two pair, either. AA, KK, JJ, and TT all beat me. There's nothing I can put UTG on that I can beat. But can I really fold?

I call one bet on the river because the pot is so big. BB calls. UTG shows his AA.

Should I have gotten away from this hand at any point? I don't think it would have been a bad call on the river against some players at this table, but UTG wouldn't have capped pre-flop without a big pocket pair or a suited AK or AQ. He wouldn't have 3-bet out of position on the flop with AK or AQ, so I had to put him on a big pocket pair. And every big pocket pair had my queens beat. Was I a fish on this hand?

Also, should I have capped on the flop, hoping to get BB out if he had overcards, or possibly slow UTG down on the turn (where I could take a free card)?