Maybe you primed it too much. This type of primer actually floods the engine temporarily.
The British used to call this tickling.
Did it run OK??

Yes, my mower ran alright after the primer bulb leaked a few drops. It's a new "plain jane" mower with a 5.5 hp Briggs. It starts on the first pull. I have no complaints with it.

Reminds me of the Amal carburetors on my '63 triumph. Push the tickler until gas gets all over your fingers.

Amal was a good and simple carburetor. I had problems when I chopped my BSA. It changed the angle and the float could not be adjusted.
So, I switched to a Mikini.
And the tickler worked great on the Amals.
Also the Amals had a clip to adjust the needle height.

No, the leaking primer bulb is not normal.