FWIW, I have similar problems. One thing I've noticed a lot is that if the overcard is an A, I can count on being behind. People just love to play Ax.

Here villian could have an A (you're behind), another pair (you're ahead) or a flush draw. When he calls your bet instead of raising, I put him on the FD. I'd bet the turn to make him pay for the FD and check behind/call the river.