Since it did the same thing with both relays, then the old one was probably good still, but if original, now 27 years old so it's best to just leave the new one in, assuming it isn't generic Chinese garbage quality.
It seems most likely that you either have a bad new fuel pump, OR bad contacts in your relay box for that relay, which you could inspect with a flashlight and spray some cleaner in, even wrap a dry piece of paper around the relay contacts one at a time and insert and remove the relay a few times trying to clean the contacts with minor abrasion, and if you can get a pick down into the socket contacts and bend a little more spring into them, that wouldn't hurt either. An inspection of the relay box will tell you if more focus is needed on it.
If the new fuel pump has a bad spot on it, might only intermittently spin up. When it won't start, you can then remove the relay and then put same relay back in to see if that helps, or even jumper the relay socket to force 12V to the pump for a couple seconds to see if it spins up.
I'd also look around to see whether there might be some wiring insulation issue, from age or rodent damage, and check the bulk wiring connectors, see this topic:
Sometimes a gremlin can be caused by the smallest thing. There are 2 large wire bundle connectors on the firewall. Both are held tight with a 10mm bolt. Any bad connection in either plug can cause just about any issue. from bad transmission shifting, bad o2 sensor readings, no start, no...
www.explorerforum.com
Ultimately, when it won't start, I'd check that 2nd gen wiring diagram for the path of the circuit and use a multimeter to trace the circuit and see where power stops, remembering that the fuel pump only runs for a moment each time the key is turned to run, if the engine isn't running. If it turns out that you get power all the way back to the fuel pump, also take note of whether the voltage drops too much when the pump tries to run, which could be a sign of frayed wiring or fouled connector(s), or of course if the fuse or relay box has fouled or weak contacts. Unfortunately many of those power checks can be a 2 person job if you just have a basic multimeter with short leads, so you need someone else to keep turning the key on and off.
Here is a portion of the fuel wiring diagram showing the most likely places to test, and the relay contact designations and wire colors: