Yes, replace the crankshaft sensor. The one your currently messing with is fine and unlikely to solve anything.

 

Any chance it could be the fuel pressure regulator? I was looking at service records on my XJS and there was an invoice where the previous owner had an issue of shut offs and it shows the fuel pressure regulator was changed.

 

My experience with fuel pressure regulators, has been when they fail, the fuel mixture usually goes very rich, to the point of belching black smoke out the exhaust pipes, and it could even flood the engine out. I don't believe you have that issue, as you said after it sits awhile it restarts, and runs fine, till it shuts down again. It's not an impossibility, as it does control fuel pressure, being a Jaguar it may operate completely different then the systems I'm used to. So take it with a grain of salt, as a suggestion. Hope it helps.

Jack