Please check out the video on YouTube:

 

I'd start with the quickest, easiest (and probably least expensive) step: Change the transmission fluid. You need 3.8 pints (call it 2 quarts). Here's the factory spec:

Motorcraft Full Synthetic Manual Transmission Fluid
P/N XT-M5-QS Specification WSD-M2C200-C

There are other products like Red Line MTL, AMSOIL MTF, Royal Purple Synchromax that would suffice.

I have no experience with Jaguar's manual transmission shifter linkage, bushings, etc. Perhaps other members here do. Do the fluid change first and evaluate.

 

SNG Barratt parts catalog (they'll send you one free) shows the gear control cables as part number C2S47848. Price on their website is $101.15, which is not bad in my book. Your cables could be stretched, resulting in those long throws.

 

I have the ford motorcarft oil for the gearbox, 2 litres of it. I will change the oil anyway.

I think I will try adjusting the cables first.

I am going to change the gaiter.