You're actually double-gutted on the flop; you turn a triple combo.
The flop call is good -- you have hidden draws that are likely to be paid if he has a hand. You don't have as much need to be super-aggressive here like you would if you only had a flush draw, which would be quite obvious if it came in.
Now the same is true on the turn, the BDNFD is well-hidden and likely to be paid if it hits and he has something decent. But now the problem is his bet size. He is representing that he has nothing or very little, so your implied odds are not as good as they may seem. Still his very little could easily be ahead of your Ace-high, so if you can get him to fold something like 99 it's a coup. So given his bet size, I raise the turn to something like $9.00, and the unfortunate & few times he pushed I'd probably call. More often he's going to fold, and sometimes he'll flatcall in which case I'm prepared to go all-in on the river if I feel he can fold a pair.