I have tried everything both ways. Without the added ground, the relay buzzes loudly. The buzzing is from the relay engaging and disengaging rapidly, and you can hear the fuel pump pulsing on and off with it.

Yesterday I did a compression test with all six spark plugs pulled out. I left the fuel system intact. After each test, I stuck a long q-tip into each spark plug hole and it came out bone dry.

 

From what we have covered so far , it appears there may be a problem with the ECA.

As you are aware, a poor connection OR low voltage to a relay will cause it to buzz. By by- passing the ECA to provide a ground to the fuel pump relay may allow the vehicle to start and run, but also is responsible for the fuel pump running all the time the key is on.
Your mission ,if you decide to accept it,,,,,,,,,,,,,sound familiar, is to do some serious checking.
We don't always have a diagram to go by , so we trace wires by color.
It is easier to trace by color than to try to figure which pin on the computer plug is involved. You may need to open some covering from a harness which makes it a lot easier .
You know which color is supposed to supply that ground, so if the connection from the plug to the relay is good , all that is left is the ECA.
Make sure the ECA has good power and grounds.

 

I only added the ground after the fuel pump staying on with vehicle off and key out issue happened AND was resolved. Also keep in mind, the fuel pump relay is hardwired to the battery in some way by the previous owner. It was working this way for quite some time without issue.

Timeline of events:

Fuel pump started staying on occasionally with vehicle off and draining the battery, requiring a jump start. Jump started it and drove vehicle to my house. Fuel pump still running nonstop with or without key in vehicle.

While trying to find the problem, I must have bumped the fuel pump relay/main relay/ECA area or some wires and the issue resolved itself somehow. (probably the relay was bad and stuck on or was getting weird power from the ECA or somewhere else)

Immediately after the last step, suddenly, the vehicle wont start at all. Cranks over fine, but does not try to start. And the fuel pump relay started buzzing whenever key is in the on position.

Replaced the fuel pump relay and bench tested the other relay.

Created my own ground and pushed the bare wire into the plug going to fuel pump relay to add some extra grounding to the existing ground wire. Fuel pump relay stopped buzzing.

Fuel pump runs with the key on whether the added ground is connected or not. But without ground it pulses constantly, making an obviously incorrect sound.

Vehicle still cranks but wont start.

Verified fuel pressure. Verified fuel flow rate. With and without the extra fuel pump relay ground.

Verified spark. Replaced spark plugs anyways. With and without the extra fuel pump relay ground.

Added a secondary 4 gauge ground from battery to frame and from frame to engine block, just in case.

It appears the fuel injectors are not pulsing which is likely cause of crank no start condition. (No fuel in cylinders after cranking, checked this with a long q-tip through 3 or 4 so the spark plug holes while doing a compression test.)

Compression numbers are not amazing, but not so bad it would cause it to not start or even rumble a little when trying to start:

150
140
130
135
100
121

Will be replacing cap, rotor, and spark plug cables soon. Probably about to pull the plug and order a new ECA.

 

I replaced the ignition control module and she finally started right up!!!

I believe the original ECM, the fuel system relay, and the ignition control module were all bad and I don't think replacing any one of them would have got her running on their own. Possibly one of them caused the others to die, but I really don't know what happened for sure. Replacing the ignition coil was probably not necessary, but it looked old as dirt and was cheap so that's fine with me.

To recap for everyone out there reading this:
Messing with and fumbling around with the relay caused the fuel pump to stop running non-stop with the engine off. I ended up replacing the relay just in case.

After this, it stopped starting up altogether. I found that I had spark and fuel, but no electrical signal to the fuel injectors.

Replacing the ECM made the fuel injectors receive electrical signal again.

After this I lost spark so the engine still wouldn't run. Not sure how this happened but maybe cranking it over and over and testing the spark plug against the engine block caused this.

I was at my wits end at this point and about to throw in the towel, but then decided to replace the ignition control module and I am glad I did.

Thanks for all the help!

 

Thank you for the update.