Bradley,
I have a 70 Road Runner and I had the same problem. My car is a California car, so it has a Holley carb. In my case, I had a leaky gasket around the accelerator pump housing, so after I stopped driving the fuel would slowly drip out of the accelerator pump until the bowl was dry. Quite the fire hazard, since it must have been doing that while I was driving!
At any rate, I realize you have a Carter and not a Holley. I would try starting the car, and after running it for a few minutes, shut it down. Then check for any obvious leaks around the carb area. If this is dry, I might suggest locking the throttle wide open. When you do this, you will squirt fuel with the accelerator pump. Let it sit for a little bit for this squirt to dry and then peer down the carb and see if you can see any fuel drips.
Or, you could just bite the bullet and take the carb off and give it a quick rebuild. That's probably what it needs anyway. :eusa_think:
Good Luck.