First a point about your bet. You don't say specifically what you bet, but it sounds like you may have gone all-in as well. That's the first mistake. Hold back a chip or two and you win if the dealer beats you both.
Now, I'll make some assumptions that complete your scenario.
We'll assume that the dealer stands on soft 17, and that you lead $10,300 to your opponent's $10,000. He is on the button and went all-in. We'll assume you matched his bet and placed a bet of $10,000 as well, holding back $300 unbet chips.
Now, the dealer has a 6 up and your opponent has a 20 while you have a 12. You should stand with your 12 here. You win if the dealer busts or makes 21. Standing gives you a 52% chance of victory.
If you hit once instead and stand on whatever total you make, your chances drop to roughly 50%. Close, but not as good.
If the dealer hits soft 17 in your game, the strategy remains the same but your chances improve to 55% for standing or 51% for hitting once.
To answer your question: Should you have stood with 13? Yes, and you should have stood with 12 as well.