Briggs - Gas Blowing Out of Carb

Briggs Engine 127802-0640-01 95091859
Carb has primer bulb
Toro Push Mower

Problem-Symptoms:
1. Engine starts and runs great for a few minutes and then runs "rough" for 15-20 seconds and then runs smooth for another minute or so and then runs rough again.
2. After troubleshooting, I noticed that the new paper air filer was wet with gas and the bottom of the air cleaner base was wet with gas.
3. With the air filter removed, the engine runs great and does not sputter or run rough.
4. With the air filter removed, I noticed a heavy mist of gas is blowing out of the carb. I believe when the filter gets wet with gas, the engine gets starved for air and then runs rough.
5. I put a known good carb in the engine and the second carb did the same thing.

Question: What causes gas to get blown out of the carb? How do I trouble shoot this?

Background.
1. The owner of the mower was mowing in very tall gas and "hit something". This caused the dipstick to get blown out of the tube spraying oil everywhere. He could not get the mower restarted and ended up breaking the pull rope.
2. I checked the flywheel and the key was not sheared. It looked fine.
3. I cleaned the carb, changed the oil, new spark plug, new rope.
4. I checked the blade and it was not damaged. I sharpened the blade and made sure it was balanced.

Thanks for your help.