You said the vibration disappeared when you pushed in the clutch; that to me would indicate driveline. A tight or worn universal joint can cause a pretty severe vibration under load and smooth out on coast.
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Our 72 V12 had a similar problem after engine temperature normalized. It would stall under load from a stop. As I recal it would vibrate a bit also just before stalling. Wait 15 or 20 minutes and it would be ok until it got up to normal temperature again.
After rebuilding carbs, fuel pumps, swapping OPUS modules, coils, etc I gave up. Then someone here helped me and led me to something I hadn’t thought of. The vacuum retard unit
The vac unit was seized and the engine couldn’t provide torque under load. Replaced it with new, all fixed.
May want to check yours by opening the distributor to view it and test it with a Mityvac.
Rich
Ignition system was replaced with that from XJS as per post#1. Should have vacuum advance now.
Once again the collective hive mind scores a success!
The reason I thought it was a dud carburetor causing the problem is because it came on suddenly, and then it didn’t start vibrating until I drove a hundred yards or so. I imagined one carb running out of fuel under load, causing the imbalance. That’s where I focused my attention until the 6 cylinder test proved me wrong.
The reason I discounted the drive train is because with the clutch in, the entire drive train is still turning. An out-of-round tire or out-of-balance driveshaft would be doing the same thing with or without the engine pushing it.
However, I had become sick of looking at the engine over and over again and decided to put the drive line theory to the test. My new house is set up with a 9000 lb 2-post lift, so I hoisted the car up and walked under it. (I’ve been waiting for over 50 years to be able to say that!) I had to take the exhaust and its heat shield off in order to even see the driveshaft, but after that the parts that I could see seemed tight. Unfortunately, I couldn’t even see the front U-joint, since the rear transmission mount and a chassis crossmember go under it, which made me wonder how may times it had been lubricated during its lifetime. I used a long rod to reach down the tunnel and press against the U-joint as many different ways as I could, and in one direction it moved and exposed a shiny bit of metal. It simply needs a new U-joint! Simple, right?
I have replaced many U-joints in my time, but I have never encountered one where they built the car around the drive shaft. In order to remove the drive shaft I need to either remove the engine or remove the rear suspension! Oh well, at least I now know exactly what the problem is and am assured that the car will once again work properly when I’m done.
Thank you one and all for your sage advice!
PS. The V12 has no fewer than 17 vacuum ports, and mine has 18 if you count the ported vacuum tap I drilled for the XJS vacuum advance distributor.
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I believe I discovered why the U-joint failed after only 50K miles. The two U-joints are not in alignment with each other. The two ends of the shaft were installed one spline off, so they were misaligned by 22.5 degrees. The car only had 27K miles when I bought it and I never had the driveshaft out, so I assume it came that way from the factory. I’ve heard that this car was built during the British Leland years, and labor was unhappy. That’s not the first factory sabotage I found on this car. The cooling fans were assembled incorrectly which made them spin backwards. I surmise the resulting overheating was the reason the original owner only put 27K miles on it on 30 years.
Kill’em, every time.
Sometimes the universals are DELIBERATELY misaligned. As on Land Rover Discovery 1 front propshaft. This from the LR manual:
Intriguing: NEVER heard of such. I wonder what the standard longevity of that is?
NEVER too old to learn yes? 
Depends. With standard suspension, much longer than the exact same UJ in a rear wheel drive vehicle. (it only does half the work in a permanent 4wd such as the Discovery 1)
With lifted suspension, half the life (or less) of the exact same UJ in a rear wheel drive vehicle.