This is an interesting question.
I think you've played right pre-flop.
I think the continuation bet is right post flop usually, but your stack size I think means you need to either push all-in or check-(call/fold).
Against one opponent I would push in this instance. Considering the second opponent here is the BB-special, I'd probably still push. It's unlikely they've hit the Queen, and you've just got to hope your caller on the button isn't sitting on high pockets or has hit the queen. Perhaps the button caller has a wideish range because they've got position and are unlikely to be re-raised in this spot.
Regarding your question about what to do on the turn, I don't think calling is a terrible option here, assuming they haven't hit the set and they've only hit the Queen, you've still got 5 cards in the deck (3 As and 2 Ts) which will be winners for you. You're paying for 23% or so odds, you're getting around 12%, which isn't horrible (that's assuming they've hit the Queen). With implied odds there's a whole lot of hands like J9s or even KT that they may have called 25% of the pot just to see what comes around on the turn, and they've pushed when they've sensed weakness.
I think AT is the worse hand you call with in this situation. I'd throw away KT (unless I had a flush draw), but AT is decent. It's just too easy here for villain to push if you're folding 2nd top pair with top kicker. I'd call unless villain generally doesn't bluff.
On second thoughts, I've forgot about the antes. With the antes, I think your preflop raise is too small. I have a habit of raising the pot pre-flop. There's no evidence I've got that this is the best way, but anything less seems to let drawing hands cheaply call. In this case, raising the pot is betting 1900. Now, it's worth thinking what you'd do if someone comes over the top with an all-in (lets say the button). The pot will contain 10775 in chips, you'll have to invest your 5875 to call. This is a hard fold to make. You only need 35% odds to call here (ignoring bubble factor). You're slightly behind a pocket Tens to Kings (30%) and you're a bit more behind Ace King-Jack (25%). On King-any other you're in front (60%), and you're still getting better than your odds against pockets lower than 9 (45%). I tend to think if you're in a situation where you're struggling to fold to a reraise, you might as well be the one that goes all in, lest your opponent has the advantage of determining the size of the pot.
With 1100 in the pot pre-flop, and a stack of 7775, you're probably in push fold territory. Push here and steal the blinds. If there was no ante, a raise to 1400 to steal 600 still gives you room to fold to a 3-bet whilst being strong enough to make the BB think seriously about calling. If they do call, you're in push/fold on the flop. Raising to only 1000 pre-flop I think makes the button assume the BB will call, so entices the button to call with drawing hands.