The engine is shut down by a small vacuum powered bellows inside the aft
end of the injection pump. Over the years the bellows wears out and
leaks - yes oil is drawn into the vacuum system.

Most likely the "shut down device" needs to be replaced - this isn't
hard to do but IS tricky as the device must be hooked onto the end of
the fuel rack inside the IP. An experienced shop will charge an hour of
labor to r/r the shut down device. It costs about $75 as I recall.

My car's symptoms were that its engine no longer instantly stopped but
rather trailed off to a stop over a few seconds. A new shut down device
fixed it.

Of course a vacuum leak in a door lock could also be the cause as no
vacuum = no shut down.