Sounds like you have the opposite of my problem, which is excessive flow to the carb chamber even when the mower isn't running. You may have a partially-clogged line from the tank to the carb. The primer bulb is helping you get fuel into the carb chamber.
Try draining the tank and checking the cleanliness of the line that is clamped from the bottom of the tank to the carb chamber. Replace and refill with FRESH gas.
May not be the ultimate fix, but this is where I'd start.
T. Easterwood, April 2007