Running out of options, here. I've replaced all the front bushes and most of the rear, including strut and sway bar etc. No more clunks or strange noises. ball joints fine, new links, tie rod ends fine, new tires and rims, and alignment. Replaced one front wheel hub. But still I have this sense of driving on ball bearings. The steering is fine and precise, but it feels like the butt of the car wants to go wherever. It's like something wants to plop from one side to the other. A good goose with the pedal and you can feel it slide over. braking will also make it shift.Tturns will and even an irregular road surface makes it feel like something is shifting from one side to the other. You can't actually see anything, it's more like a feel that needs only the tiniest correction with the wheel, but it's a very strange feeling, like the wheels flopping over from a negative toe in to a positive one or something. Any suggestions?

Hi what type tyres are you running it can make abig difference together with tyre pressures.
I had similar feeling on other cars especially rear, turned out my pressure gauge was 10 PSI out what I thought was 30 was in fact 40?

I was never a fan of the recommended Pirreli P6000 for what its worth.

Tires new, pressure correct. Low profiles on 19" rims.

Rear end wheel alignment?

What brand of tire? Garbage tires like Triangle, Riken, Nexen, etc. can give a very unstable feeling, especially if its a 35 series or lower profile.

Will be even lower profile on 19" I expect

They're way out of spec for size for the early car; the suspension will never be right with 19".

Are they even Jaguar S-Type rims? (They can't be right even if they are.)

What was it like on factory rims?

I'll have to check the brand as I don't recall. They were $150/per on sale. But with lower profile tires like these you should get less sway, not more. And I'm sure it's something mechanical; it's not a general loose feeling but an actually lurch from side to side. All wheels were aligned. It's greatly improved with all the components replaced but you still have to struggle with it at high speeds. I've never driven the car without the existing rims and tires.

Ditto + rear toe adjustment.

-Did you replace the rear toe/tie rods and not get them aligned correctly?
-Did you by chance replace the lower control arm bushings with Delrin bushes instead of the stock ones?

Rear wheels were aligned after I replaced the tie rods. Stock bushes.

The only other thing I can think besides tires is that when you put your control arms back on, some misalignment was introduced that changes the path the control arms follow as they move up and down. this could cause the tires to move side-to-side.
Alignment is done with the suspension in its neutral position. Perhaps the wheels are properly aligned just sitting there, but when the suspension moves up and down, the path the control arm follows moves it out of alignment.

Do you see any indication of tire scrub? Is there any fine tire powder sticking to the bottom of the car or bumper?

The S-type rear suspension is a Semi-Trailing arm design. If one of the lower control arm bushings is not centered in the A-arm properly, the path the control arm follows could cause the problem you are seeing.
_____________
From the internet:

Semi-trailing arm suspension has the trailing arm pivoted at inclined angles - about 50 to 70 degrees. Otherwise are the same as trailing arm suspension. Apparently, the semi-trailing arms are half trailing and half transverse. You can analyse it by splitting it into two vectors, one is the trailing component and another is the transverse component. The trailing component leads to understeer, as already mentioned. On the other hand, the transverse component is actually equals to a swing axle suspension. Now, you may remember that the swing axle suspension always introduce oversteer due to body roll. As a result, the two components cancel each other and result in near neutral steering response.

Semi-trailing has a disadvantage - when the wheel moves up and down, camber angle changes, unlike double wishbones suspension.

No matter semi-trailing arm or pure trailing arm suspensions, since they are rigidly attached to the wheels, inevitably more shock and noise could be transferred to the car body, especially under hard cornering or running on bumpy roads. Moreover, a lot of unsprung weight of the trailing arm leads to poorer ride quality. Therefore most modern sedans replace it with multi-link or double wishbones suspension. Trailing arm / Semi-trailing is disappearing in the industry

I'll check that out. It feels a lot like when the bushes to the control rods were blown, allowing the wheels to toe in and out like 1/2 inch. They essentially tended to steer the rear wheels independent to the front. Replacing them improved it along with all the other parts replaced, but the wobble is still there. And it seems more like inertia flops whatever is moving from side to side; even when the car is level, it will flop over when going around a curve, requiring a slight correction with the wheel.

I had the same issue; everything tight or new but still the looseness feeling especially at highway speeds. It turned out to be rear left wheel bearing. I was shocked when the tech rocked the wheels while car was up in the air.

I was charged around $360 - bearing + R&R + alignment.

When I replaced all these components, I did so after receiving a list of what I needed from a suspension shop, which did include a loose front wheel hub (their estimate was well over 6 grand; I was able to do it for less than a grand). I would be surprised if they missed a loose rear wheel, and doubly so when I had all 4 wheels aligned after the work was done. This is the part that has me scratching my head; I've grabbed all 4 wheels on jackstands and never found any play, yet it's there. Maybe I'm going to have to take it back to the shop and have the pros look at it again.