Check for sparkly bits in the diff oil when you drain it.

@1988smithy had a diff issue on his old L320 wonder what his symptoms were?

J
L319 but tomato tomato 😉

Front diff carrier bearings, particularly nearside. No whine but quite pronounced rumble on right hand corners loading up nearside wheel. I changed wheel bearings first as it was very similar. Oil was good, bearing races were all pitted.

R.e OP's issue, any mentioned wheel bearing yet?
when the car isnt up to temperature i always run it couple of mins before moving off every morning or when its been sat and cooled say if im sat in slow moving traffic if i slightly come off the brake whilst its in drive the car will shake a bit like someone is pushing it then pulling it back like a shudder soon as i touch the Accelerator to drive away its instantly smooth when its up to full temperature its doesnt do it
Does it feel drivetrain or engine shuddering?
 
So when im cruising there like a constant hum whine droning noise but only whilst the Accelerator is pressed one you let off the gas its silent soon as you touch the gas again any amount the noise is back instantly
Sounds like diff, but worn gears rather than carrier bearings.

I think early front diffs had some mad coating inside that broke up and contaminated the oil.

Drop the oil
 
If you check bearings they can hide play quite well on these

Take wheel and pads out, then you can feel roughness when you spin the hub
 

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