"I would like to know how better players would have played this"
Reluctantly I reply, as I don't meet the requirements of the above phrase. None the less, I do have an opinion, however uninformed. lol
OUts against are: any spade = 9,
any 10, or J or Q = 9 more (we already counted them in spades),
any 5, 6 or 7 = 9 more
any 9 = 2 more
our sub total here is subtotl= 29
Unknown outs against include the third and fourth of any over pairs. ie 10,10 = 2 more 10's out against you.
29+ outs out of 47 cards is a fine incentive to take the pot now.
However, I have many times lost similar pots to the pocket 9,9. Or in this case the pocket AA on the turn. Therefore, bet the pot, not your entire stack. At these buy ins a $12 bet is very big, and as your bet illustrates, you can't always bet people out of the pot regardless of the size of your bet.
Can't wait for the results.