"moretoys" is right that you should have two 12 volt sources at the shutdown relay. One hot all the time and the other will be hot with the key on as was stated.
The PCM will turn the ASD and the fuel pump relay on for about a second when the key is turned on. It will work both the ASD and Fuel pump relays as they share the same ground provided by the PCM. You can see if look at the wire diagrahm and see that the ASD and fuel pump relay ground wires splice together and go to one pin at the PCM. The PCM must see the Crank sensor and Cam sensor signals to turn the ASD on and keep it on once the eng is cranked.
A scanner really makes this so much easier as you can have the scanner comand the PCM to turn the relays on to test them but without a scanner you have to do it the hard way. You can check for power at the injectors while you crank to see if the relay is working. But if its not then you will need to check the basic componets that cause the ASD relay to turn on like the crank and cam sensors.
I like to moniter the ground circuit for the ASD and fuel pump relays. You can just use a digital voltmeter and tap into the asd ground. When you crank the eng it should go to right about zero volts but if it stays near 12 volts then the PCM is not grounding the relay. You could even have somone ground the asd ground for a second while cranking to see what it does. I have done this before but was using a scanner as it told me more info when I supplied the ground myself to se what it did. Some engines will start without a cam sensor signal if the eng is cranked long enough. The PCM will start trying different fireing orders to work the injectors if it does not see the cam sensor but all else works. But you need to see just what the ASD is doing. Thats why if you tap a voltmeter into the ASD ground that the PCM supplies you can watch the volts.
When you turn the key on the volts should go to 0 for a second and then to 12 as that is the relay turning off. Then when you crank it should go to 0 if the ASD relay turns on. If it stays at 12 while cranking then you have to figure out why the PCM is not grounding it.
Do you have any scanner at all to use while you are working on it. Ron