Don't forget that the ASD relay will only send power to the coil for a second or two before it turns off if the ECU determines there could be a problem. It's just like when you turn the ignition to "run" but don't actually crank the motor. You'll hear the fuel pump run and then turn off. It won't turn on again unless the ECU is convinced that the motor is actually turning over. If the crank or cam sensors are not signalling properly that could cause the fuel pump and ASD relays to turn back off as soon as you turn the key to "run" position.

Do you have power to the coil for a second or two right after turning the key to "run"?

Do you hear the relays clicking on and off?