mac here ,your 300ZX , similar to my 93 J30 (VG30DE motor too I believe) yet I don't believe the fuel pump supposed to overe-run, AFTER engine Completely off ?

Maybe that's the design , but in all my time I've never seen or heard - it's sooooo counterintuitive , that a FUEL pump will still run - eng. off + key out FOR ANY length of time-out but I'll never claim GURU status just an "IDIOTUS GENERALIS" Diploma

Seriously I just worked on mine last October and heres how INFINITI / Nissan probably does it in your case too = power from a relay usually @ front of vehicle cotrolled by the ecm (which I believe always disconnects from power when car off - to protect from bad jumps and bad Battery chgr's) .Relay sends power BACK to a "Fuel Control Module" (FCM has "relay power", ECM sig.,FCM OUT to F/pump, lastly a ground connection, note:and pump goes series w/ a HEAVY DUTY "Power Resistor" to gnd. too).ECM sig. turns on pump via module at "right times" but ALSO is responsible for SHUTTING DOWN w/no RPM condition ,other engineered-infault response and obviously eng-Off or after the 5 sec. prime etc.
Unless something like a"Mod" or weird wiring situation/mishap is allowing SOME part of this ckt. to kick in?
Another thing is that if your F/pump is accessable back there in the trunk cmptmt. they're easy to R&R, prett yreliable ,IT IS PUMPING , cheap but messy and sort-a dangerous to mess with gas

!, you don't want to go there yet !
Now the power resistor is SO big and kick-AZZZ that unless it smells burnt , and/or looks somewhat fried but most importantly if you can test Ohms w/multimeter ( FSM gives spec. I believe mine was .7 ohm? seven tenths !

) -- You should now be suspecting that Module

and FSM i believe HAS a basic "Field Chk." to try.
Lastly maybe i'm learning something here , but the fact a F/pump OVER-RUNS after - again COMPLETE SHUTDOWN , seems to be the heart of this weird fault . GT 2/22/2018