I HAVE A 1984 MERCURY GRAND MARQUIS. I TURN THE KEY ON AND IT WILL GIVE IT A SHOT OF GAS AND STOPS. IF I HOLD THE KEY ON THE ON POSITION IT KEEPS GETTING FUEL BUT ONCE I LET GO IT STALL. IT HAS A NEW FUEL PUMP AND I CAN HEAR IT RUN WHEN I TURN THE KEY ON. HOPE YOU CAN GIVE ME AN IDEA WHAT TO LOOK FOR. THANKS MARIO

Replace fuel pump relay for sure (check owner's manual, or google it), Check inertial fuel shut off in L trunk. There are some other things, someone will probably chime in

Also check fuel fiter not sure if a 1984 is fuel injected or carbureted if fuel injected filter is down underneath car on frame rail by right rear tire, if carburated it is either in carburetor by gas line or in the the gas line just before carburetor. It must be fuel injected because you you could hear the pump running in tank fuel injected cars have there fuel puns in the Gas Tank

I'm also having the exact same issue with my car the switch in the trunk is on the drivers side and when you remove it give it a good shake if it rattles it's bad I found out that you cannot bypass this button if that's not the case I would remove the wire on the fuel pump itself and connect it to a twelve volt battery if you don't hear a buzz sound or a sound at all the pump is bad this is what my car looks like at the moment

Could be ignition switch, they go bad. Fuel pump only runs a few seconds, then shuts off if the car hasn't started.