I took the carb apart, and it didn't have any dirt or sludge in the bowl, so I gave it a quick clean with mineral spirits (didn't have carb cleaner) and replaced the float valve and seat then reassembled it with new gaskets. I put in a new filter element and mowed half of my yard before I noticed that the gas was dripping from the air cleaner again.
Just an observation, but the little nib in the center of the primer bulb has a hole in it. I'm not sure whether this is normal or even if not whether it is relevant to the problem. I've had it suggested to me that perhaps the intake valve isn't seating properly, such that the airflow in the carb throat might momentarily reverse during the compression stroke. Any ideas?