It sounds like you're having trouble with the start/stop solenoid.
The solenoid is located on the firewall side of the pump, about 1/2 way down and just above were the injector lines connect.
The solenoid has two wires with spade connectors on them.
One wire is blue, the other is black.
The only thing you can do is remove the solenoid and check the spring loaded plunger operation.
The plunger has a black rubber tip on it and that is one item that can fail. Normally you'll see a piece or two missing from it. That means the thing is not doing it's job.
To test the solenoid itself, you need to pull the solenoid body, it's internal spring (like a ballpoint pen spring only tougher) the metal cylinder and rubber tipped part out.
Put the parts back into the solenoid once it's all out, and make sure you have a protective grip on the plunger end. Apply 12V to the metal post on the solenoid and see if the plunger pops in and out cleanly.
If it works ok and, there is no damage to the rubber tip then you're problem may be in the signal voltage from the second wire.
One wire is "trigger" voltage to open the solenoid and, the second wire is "maintainanence" voltage to hold it open.
With the key "ON" check each for power. One should be active while the other is dormant (I believe, correct me if I'm wrong here guys) With the engine running check if the second wire is active.
That should get you started..... on the road to diagnostics/fixing...
pastor bob..... ps: I really suck at electrical things...