You're playing two pair on a coordinated board way too slow. This flop is just too dangerous to let anyone see a relatively cheap turn. Hell , on any flop I play it fast, especially with bottom two. Two pair on the flop is typically a strong hand, but never the nuts. With bottom two you have the additional worry of being conterfeited. I wouldn't even play a set this slow on a flop this scary. Raising not only serves to put more money in the pot, but also helps define your hand. I don't feel you were betting enough to define your hand or your opponents.

If it was me I would definately bet the pot on the flop. I'm not a big fan of overbetting unless you want to give the impression of bluffing when your not, but I might even slightly overbet the pot here. When the action came back to you on the flop and you knew you would be HU I would probably push. Unless another ace came on the turn I know I would likely be betting enough to commit him to the hand anyways. I also know that if I bet a significant amount on the turn that the range of hands that are calling have a good chance of being ahead and that if I am re-raised then I'm pretty much committed to calling myself. I'd rather put all my money in on the flop when he hasn't had that chance to see another card. If I happen to be beat on the flop then I'm most likely still going to be beat on the turn. If that happens, oh well. If his stack was deeper though then I would play it more cautiously.