Fuel pump issue
The usual problems are leaking injectors, check oil for gas smell. Or the plastic fuel pump "pulsator" that fits between in tank fuel line to fuel pump, most replace this with a rubber hose and hose clamps if its leaking. Otherwise fuel pump could be not holding pressure.
Don't forget the fuel pressure regulator, if the diaphragm in the regulator has split it can leak fuel via the vacuum hose into engine ... Again smell the oil for gas smell, any oil dilution can be a problem. even though gas will go from liquid to vapor very quickly a lot will cause oil dilution problems so keep an eye on oil pressure.
In regards to the pump staying running for a few seconds after shutting engine off is not normal, would have to confirm via wiring diagram however the oil pressure is used as an indictor the engine is running to allow fuel relay to energize. However when ignition is switched off the fuel pump should stop right away, some early years also use the ignition reference to enable fuel however this is just from memory.
In short the fuel pressure dropping over minutes or hours would not worry me, if its instant a lot of fuel pressure loss quickly would mean its going into the engine if injectors not holding the pressure would go into the oil via gravity.
Good luck










