Weak top pair and then an overcard turns

This comes up often enough to be a leak. I have no idea how to play this. The villian is your typical 40/5/1 3/6 6-max calling station.

The problem is overcards turning in general, but the Ace in particular. At the turn, if I bet and I'm ahead, he'll usually fold. If I'm behind, he simply calls. If I check, he'll bet if he's ahead and check if he's behind, which really sucks.

If he simply calls, then I'm stuck in a tough river situation. Check/calling seems weak. Check/folding seems completely lame. Betting seems foolish as the odds of me being beat seem better than 50%.

How do people play this?

PokerStars 3/6 Hold'em (6 handed) converter

Preflop: Hero is BB with Q[img]/images/graemlins/diamond.gif[/img], 7[img]/images/graemlins/spade.gif[/img].
UTG calls, <font color="#666666">2 folds</font>, Button calls, SB completes, Hero checks.

Flop: (4 SB) 7[img]/images/graemlins/diamond.gif[/img], 3[img]/images/graemlins/diamond.gif[/img], 3[img]/images/graemlins/spade.gif[/img] <font color="#0000FF">(4 players)</font>
SB checks, <font color="#CC3333">Hero bets</font>, UTG calls, Button folds, SB calls.

Turn: (3.50 BB) A[img]/images/graemlins/heart.gif[/img] <font color="#0000FF">(3 players)</font>
SB checks, <font color="#CC3333">Hero bets</font>, UTG calls, SB folds.

River: (5.50 BB) J[img]/images/graemlins/spade.gif[/img] <font color="#0000FF">(2 players)</font>
<font color="#CC3333">Hero bets</font>, UTG calls.

Final Pot: 7.50 BB