Just to be clear, the carb will draw fuel in when it's running, and you could be seeing that. I'd recommend pulling the plugs, turning the car to 'run', and pulling the throttle cable while looking down the primary barrels. Without the engine running, you should see the Accelerator Pump shooting fuel into the carb. Once you're done, turn the engine over a few times, put your plugs back in along with the wires, and crank her back up.
The only reason to pull the plugs is to avoid flooding. There's a good chance you'll flood it while playing with the AP, and it's a lot easier to restart if you'd already removed your plugs and they're still dry.
The AP is the likely culprit. Another possibility is that the car has mechanical secondaries (not stock, but a common mod) which you're not used to, and it'll make the car bog if you drop the throttle too quick. While you're checking the Accelerator Pump again, also pay attention to the two larger bores. If you slowly turn the primary throttle linkage (or pull the cable) and the secondaries (larger bores) open smoothly with the primaries, then you have manual secondaries. Let us know what you find.