The car stalls at random times. Sometimes while driving and sometimes when I've finished driving and I put it in park and it stalls before touching the key. I've read on other sites that the fuel pump is likely since the car does start up again right away and the dash board lights stay on when it stalls.

SOURCE: Car keeps stalling when comes to a stop

check vacumn hoses? egr valve,pc valve

SOURCE: Random stalling, then won't start

was fuel filter replaced? it also could be a bad coil pack

SOURCE: 2000 blazer 4.3 hard to start (fuel problem)

Do you have the fuel tester hooked the in the overgrown air valve on the fuel rail in the back of the engine? Or do you have the fuel line unhooked in the back and hooked to the pressure pump? You want the fuel pressure to be in that 60 lb range so if you are hooked into the pressure valve on the fuel rail allowing fuel to go through, then I am betting it is the fuel pressure regulator. To get to that, you have to take the throttle body off and take the plastic plenum that covers the injector packs. On the back of the mount where your six pack of injectors are sits the pressure regulator. If it is faulty then it is dumping fuel instead of holding the pressure like it should.
Get a Chilton or Clymer manual and it shows you in pretty good detail how to fix it. I've done it to my 98 S10 with the same engine.

If you unhooked the fuel line so you are losing pressure from the line back to the pump then I would say your pump is letting fuel back to the tank when it shuts off. My money is on the first possibility though.

Good luck

SOURCE: 92 Chevy Suburban Electrical Fuel System

Check fuel flter at fuel pick-up intank (looks like a small wire sock). Remove fuel filter and fitting at fuel rail. Blow out fuel lines w/about 35lbs air press. Ck electrical @ relay - should be 4 connectors - one is key on hot/one is always hot/one is ground and last is hot to fuel pump. Good Luck

SOURCE: 04 Chevy Cavalier fuel pump

The fuel pump relay is buzzing.the possibilities are:--- 1) the relay itself is faulty. 2) The relay is not connected properly. 3) the relay ground wire is connected to ECU (electronic control unit also known as cars computer). -------- You say relay is just replaced,so relay is good.Check the relay connection,remove the relay clean the relay and reconnect it.Then see if same problem is noticed like relay buzzing,then in that case the relay ground wire connection has to be checked.The ground wire is either loose or there is cut in between the ground wire. At the fuel pump connector, the gray wire is for the fuel pump. The fuel pump relay will energize for 2 seconds each time the key is turned on. Test for power on the gray wire by turning the key on, wire should power up for 2 seconds.Test at the fuel pump connector under the car at the fuel tank. Touch the voltmeter red lead to the thicker gray wire. Touch the black lead to a black wire or chassis ground (bare metal). Turn the key on and you should have 12 volts for 2 seconds.A-black wire is ground. C-gray wire is fuel pump power feed. Measure voltage at these two terminals when the key is turned on.If there is no voltage then by pass the fuel relay by jumper.Pull relay and look at the bottom of the relay. Find the terminals labeled 30 and 87. Jumper the corresponding terminals together to bypass relay.You will see pins 30,85,86 and 87. 85 and 86 are power and ground to the relay coil (turns the relay on). 30 and 87 are the switch contacts within the relay (main power comes into and out of the relay to the fuel pump on these terminals) Placing a jumper from 30 to 87 should supply power to the fuel pump.
------------ Now this is the main test result:--- If pump did not run with voltage applied, you will need to replace the fuel pump.By bypassing the relay,then in that case the fuel pump is faulty and that is causing the problem.Pin 86 gets its ground from the PCM (computer) It will only get a voltage for 2 seconds when the key is turned on, will have a ground full time while cranking the engine or with the engine running.But if pin 86 has voltage to it forever,then ground is short.It should have voltage only for 2 seconds when you crank the car/engine. ------------- But if all checked out ok then it can ECU fault.The ECU itself needs to be scanned and checked. ECU/PCM/ECM are one and the same thing. ----------------- This troubleshooting will help. Thanks.Helpmech.