I have a no start problem ,

Find another relay with an identical part number(maybe the fuel pump relay)& swap it in place of the suspect EEC relay.See if the buzzing quits.If it does,the EEC relay is bad.There might be a ac WOT cutout relay or fan relay with an identical part number that you can use instead of the fuel pump relay.The fuel pump relay & EEC relay will both have to be in place in order for the engine to start.A relay should click when you turn the key on,but it shouldn't make a buzzing noise.BTW-the EEC relay is energized when you turn the key on and then it supplies ignition on power to the computer,injectors,solenoids and fuel pump relay.Therefore, if the EEC relay is defective, these components are Not going to work.If you don't hear the fuel pump prime when the key is first turned on,that's a sign the EEC relay is bad.One of the injector wires should have battery voltage, if the EEC relay is powering up.If your vehicle doesn't have a matching relay to the EEC relay(except for the fuel pump relay)just pull the FP relay out and slide it into place of the suspect EEC relay for testing purposes.If the buzzing noise stops,then replace the relay because there's no way to test if the car will start without having both relays in place.

depending on the vehicle , most domestic's have an eec relay which also have a fuel pump relay that are energized by the ignition switch. e e c stands for electronic engine controls. no worry's unless they don't come on! on fords the eec feeds power to the fuel pump relay also .

If there is no voltage at the ignition coil then you need to check and see if you have voltage at the EEC power relay which supplies voltage to the ignition coil and the fuel pump relay. The ignition switch applies voltage to the EEC power relay. The EEC power relay and Fuel pump relays are found together usually on the drivers side fender apron, about half way between the headlight and the firewall

This should be in the owner's manual. I found the below on autozone.com, their repair help location:
An inertia switch is used as a safety device in the fuel system. The inertia switch is located in the cab, generally under the dashboard on the right side. It is designed to open the fuel pump power circuit in the event of a collision. The switch is reset by pushing each of 2 buttons on the switch simultaneously (some models use switches with only one reset button). The inertia switch should not be reset until the fuel system has been inspected for damage or leaks.
When the ignition switch is ON , it turns the EEC power relay ON . The EEC power relay provides power to the powertrain control module (PCM) and the control side of the fuel pump relay. Power for the fuel pump(s) is supplied through a fuse link or high current fuse attached to the starter solenoid (battery side). From the fuse link or high current fuse, current flows through the fuel pump relay and inertia switch to the fuel pump(s). The fuel pump relay is controlled by the PCM.
When the ignition switch is turned ON , the fuel pump(s) will operate. If the ignition switch is not turned to the START position the PCM will shut the fuel pump(s) OFF after 1 second. The PCM will operate the fuel pump(s) operate the fuel pump(s) when the ignition switch is turn to START position to provide fuel while cranking.
After the engine starts, the PCM will continue to operate the fuel pump(s) unless the engine stops, drops below 120 rpm or the inertia switch is tripped.

I would check the EEC relay, and the fuel pump relay too. But it sounds like the ignition module is loosing power, or the pump is not getting power after the engine starts.
The computer must see oil pressure or RPM to power the pump relay after the engine starts.

For 1999 Buick LeSabre - but you took #1 off the list, so here's the whole "failure to start" list:

Priority Action Part Type Cause
1 Inspect Battery Battery Discharged or Faulty.
2 Inspect Battery Cable Corroded, Broken, Shorted or Poorly Connected Battery Cable.
3 Inspect Distributor Cap Distributor Cap Cracked or Burned.
3 Inspect Fuel Filter Clogged or Dirty Fuel Filter.
5 Inspect Ignition Coil Faulty Ignition Coil.
6 Inspect Wireset Worn, Damaged or Faulty Spark Plug Wire(s).
7 Inspect Spark Plug Incorrectly Gapped or Fouled Spark Plug(s).
8 Inspect Starter Starter Motor Faulty.
9 Inspect Fuel Pump Low Fuel Pressure.
10 Inspect Starter Relay Faulty Starter Starter Relay
11 Inspect Fuel Pump Relay Faulty Fuel Pump Relay.
12 Inspect Fuel Injector Pressure Regulator Faulty Fuel Injector Pressure Regulator.
13 Inspect Ignition Switch Improperly Connected or Faulty Ignition Switch.
14 Inspect Starter Motor Faulty Starter Motor
15 Inspect Fuel Injector Dirty or Worn Fuel Injectors.
16 Inspect Fuel Pump Strainer Clogged or Dirty Fuel Pump Strainer
17 Inspect Engine Control Computer Incorrect Operating Information Being Delivered and Sent From The EEC.
18 Inspect Fuel Tank Fuel Tank Empty.
19 Inspect Carburetor Carburetor Flooded or Faulty Choke.
20 Inspect Distributor Rotor Distributor Rotor broken, burned contacts, or loose
21 Inspect Distributor Worn, Loose, or Incorrectly Installed Distributor.

sounds like a blown fuse look at the connections at the starter relay that the battery cable connects to these wires the connect to the bolt on the relay that the battery cable conects to are fuseable links and you describe one of them as blown. You need to look for power with a test light to find the power and the ightion switch is a high failure rate part on this car and they were involved in a recall many years ago. thier so old now as you just chage them as Ford doesn't houner the recall any more the main relays also rust out at their connectors and the relays themselfs go bad too.

The ignition switch powers the EEC relay, and the relay turns on the EEC which runs the fuel pump relay. So I would replace both as both have a lot of problems. As for a diagram, you will need to go to AutoZone and I think they can print one out for you on their ALLDATA program.

There is a fuse for the fuel on the firewall inside the cab:




Fuel Pump The fuel delivery system uses either a high or low-pressure in-line or in-tank electric fuel pump, with some models equipped with both. It is a recirculating system that delivers fuel to a pressure regulating valve in the throttle body and returns excess fuel from the throttle body regulator back to the fuel tank. The electrical system uses two types of control relays, one controlled by a vacuum switch and the other controlled by the powertrain control module (PCM) to provide power to the fuel pump under various operating conditions.

An inertia switch is used as a safety device in the fuel system. The inertia switch is located in the cab, generally under the dashboard on the right side. It is designed to open the fuel pump power circuit in the event of a collision. The switch is reset by pushing each of 2 buttons on the switch simultaneously (some models use switches with only one reset button). The inertia switch should not be reset until the fuel system has been inspected for damage or leaks.
When the ignition switch is ON, it turns the EEC power relay ON. The EEC power relay provides power to the powertrain control module (PCM) and the control side of the fuel pump relay. Power for the fuel pump(s) is supplied through a fuse link or high current fuse attached to the starter solenoid (battery side). From the fuse link or high current fuse, current flows through the fuel pump relay and inertia switch to the fuel pump(s). The fuel pump relay is controlled by the PCM.
When the ignition switch is turned ON, the fuel pump(s) will operate. If the ignition switch is not turned to the START position the PCM will shut the fuel pump(s) OFF after 1 second. The PCM will operate the fuel pump(s) operate the fuel pump(s) when the ignition switch is turn to START position to provide fuel while cranking.
After the engine starts, the PCM will continue to operate the fuel pump(s) unless the engine stops, drops below 120 rpm or the inertia switch is tripped.

We are having similar problems. My husband replaced the fuel pump, fuel regulator and now it cranks but dies when you give it a lot of gas. He thinks the fuel injectors are stopped up. I read where one guy replaced the computer and fixed it and another one replaced the EEC. Check the relay contact and clean and check all grounding wires.