Hello,
I don't have a suggestion for the repair but have a hint.
There is a website/forum for the Ford/Lincoln LS, which is basically the
same as the Jaguar S-Type. Sheet metal is different but it appears the
engine, drivetrain, suspension and most everything else is all Jag.
https://www.lincolnvscadillac.com/forum/forums/lincoln-ls.7/
There is also a section for servicing, including some manuals.
I beleive yours would be the same as a Gen 1, six cylinder.
There are a few regulars on there that seem to eat, drink and know pretty
much everything to know about the LS. They may be able to suggest a starting
point.
I don't want to mislead you but the fuel pump is a likely suspect, I've seen
enough posts about replacing it where it looks like it's something that
doesn't last the life of the rest of the car.
Good luck.
-bruce
b...@ripco.com