"Probably the ECM" doesn't sound like a competent shop's answer to me. There is an answer if you follow the proper troubleshooting procedure & have the proper tools. This will need a capable scan tool to diagnose the problem.
Always diagnose first. You may want to reinstall the old parts & start over.
The ASD relay will not stay on with the key on if there is no crankshaft position sensor signal (engine not turning). It will come on when the key is first turned on for a second or so.
Do you hear the fuel pump 'hum' in the rear for a second or so when the key is first turned to 'run'?
I borrowed this from Allan's post. Pins #1 & #2 from the battery are hot all the time. When the PCM grounds #5, the relay will close the contact & current will flow to #4.
The PCM will lift the ground from #5 to turn the relay off.