Hi, welcome to JA.
Running the pump continuously can damage it when the ignition is just in the ON position...the EFI relay should not be getting hot, and feeling relays click does not confirm that they are operational. Do you have access to a voltmeter for testing? You'll need to test the EFI and Circuit Opening Relays at the wires themselves under the relay box using the wiring schematic above.
The EFI relay has (4) wires: with the key in the ON position and the relay installed, the Black/Yellow, Black/Green, and White/Red wires should have power (around 12 volts)...the White/Black wire is ground for the relay. If the relay "clicks" more than likely the Black/Yellow has power and the White/Black has ground - these are the electromagnet part of the relay...but if the Black/Green has no power the EFI fuse or wiring is bad, if the White/Red has no power, the EFI relay is bad...the cicuit opening relay's location is posted below..
Go to the circuit opening relay, clean the wires off well so you can see the colors, can you list all the wire colors for me? Also, testing with a 12 volts test lamp can be inaccurate, especially when dealing with relays, but since that EFI relay is grounded and not triggerd by ground the test lamp is alright to use there.
The circuit opening relay should be under the hood, against the firewall on the passenger side...is this where your looking? Not in the passenger compartment inside the cab...
Sorry I just double checked, the illustration may be incorrect...see the info below and look under the passenger kick panel where the right foot rests...
PURPOSE The circuit opening relay is responsible for providing power to the fuel pump.LOCATION Located behind the right kick panel.OPERATION Power is applied to the STA terminal of the relay from the starter when the ignition switch signals cranking mode. This allows the fuel pump to operate until the engine starts and runs under its own power. A signal from the Air Flow Meter (FC), keeps the relay closed until the ignition switch is turned OFF and de-energizes the EFI Main Relay. The fuel pump operation is then interrupted.
Sorry, I left a post above after I said it was under the hood...have you seen the second location?
No, just becuase the EFI is supplying power to the Circuit Opening Relay, does NOT mean the COR is powering the fuel pump...the COR sends power to the fuel pump via a Blue wire, have you found this relay under the passenger kick panel as I corrected earnlier?
No not yet, did you look under the KICK PANEL? This is the side panel where the passenger's right foot would rest, hence "kick panel." The relay should be under there according to the location I found afterward. We need to find this relay before testing and should not probe any blue wires until we know it is from this relay, especially if your using a test light.
No running the truck out of fuel will cause a weak fuel pump to go out, but not the COR...the COR is common to fail on it's own. Below is the cost and part number at the dealer...i think you can order one from AutoZone as well, if you have one in your area...you may be better off going down to a local junk yard and use the part number below to get a used one for about $5-10, since they're not too cheap. I'll see if I can find any other location illustrations, but from all my information, it should be near the enigne computer which you've already found.
No problem, let me know if you still have difficulty finding that relay or replacing it. Happy Holidays if I don't here from you...