As Jerry stated, if you have not replaced the fuel pump diaphragm yet, you really need to. In my earlier post, I noted that these pumps are just barely adequate for the application if they are in good condition. Any stiff diaphragm or leak at the pulse line or as Dave stated, leaking at the check valves will cause just what you have been experiencing. Obviously, one check valve allows fuel in to the pump from the inlet hose leading from the tank. It blocks the flow back to the tank, so when the diaphragm moves, it pushes the fuel to the carburetor. Conversely, the other check valve will prevent flow from the outlet side that leads to the carburetor. This allows the diaphragm to create a low pressure (vacuum) area inside the pump to pull fuel from the tank. Leaking check valves can be caused by debris or corrosion. Clean them up and verify that they are working before reassembly. Again, you have pretty much tracked down your issue to this area by pumping the bulb and having your issues go away.

Steve