It could be worn float valve or merely debris between the valve and seat.Most of the time it is simply debris on these. Possible float saturation (gas in it) tho not likely. In either case ensure that the fuel tank is clean, same with the fuel line or replace it when you service the carb or it can happen again. With a watchful eye, and a bit of mechanical aptitude- you could tend to this yourself. If unsure, consider having it done or simply replace the carb and clean the rest.

Unless I'm missing something, you said you "pressed the primer bulb and then saw gas streaming into the air filter compartment", that's how the primer system works. You press the primer bulb and it forces gas through the jets to the carb throat. I would check to see if gas is leaking into the "air filter compartment" without pressing the primer bulb. If, after mowing, you take the air filter off and just let the engine sit there and still see raw fuel running into the air filter then you have a problem with the needle and seat as mentioned above. If not, then you probably don't have a fuel leak there.

I disassembled and reassembled the needle valve, and all seems well now. I mowed the lawn a few hours ago and put the mower back in the garage. Neither the garage nor the house reeks of gasoline. Thanks for the input. I always like these learning experiences.