OK that helps.
The diagram you found is not for 1998 as you describe it. There are free year specific FSMs in pdf on the net. The ASD is not connected to the alternator field wire in 1998.
It is normal for the ASD relay (& fuel pump relay) to only stay energized for 2-3 seconds after the key is turned on unless the PCM sees a crank sensor signal. So unless you are cranking the engine or it is running the relay will not have output to the injectors or coil after than time.
The path for 12v from the energized ASD relay for 1998 is through fuse 21, also in the PDC, then to the coil & injectors as well as the O2 sensor heaters. So check fuse 21.
If that fuse is good here is how to check this circuit. Use the diagrams of the relay socket terminals below to check the input & output or the ASD relay.
Pull the relay. Check terminal 1 for 12v power (from fuse 6). If nothing check fuse 6.
Next make a short jumper wire & jump from terminal 1 over to terminal 2 (see the red circled #s). That should send 12v power to fuse 21, then to the coil & injectors. If yes, see if the engine starts.
Report findings for more info if needed.