Twitching

[wezdavo]

If it's the rear of the bike that feels peculiar then perhaps that's where the problem is!
Have you checked things like worn swingarm bearings giving lateral play, or excess slop in the vertical direction caused by knackered bearings in the shock linkage? (Note there will be 2 or 3mm vertical play at the rear wheel even with new bearings in the shock linkage, more than that indicates a problem.)
The OE shock was never brilliant to start with and if there's a fair mileage on the bike, particularly if it's spent a good proportion of that heavily laden, then it may be worth trying a replacement.

If the front suspension was working properly i would agree but as he said the fork is leaking oil, so that makes thing a bit more complex..

If he's losing oil, the forks are losing dampening, meaning they are returning too quickly, which will upset the bike while cornering, even worse if only one leg is leaking as it will twist the front end..

If it were me i would get the forks serviced, and as Fazerider says check the rear shock (best to do with the shock off and the spring out) or you can do it car style by pushing on the back end and letting it go watching for a smooth and positive stop at the top of the stroke, check linkage and arm bearings.

Set sag and go for a ride, if its still not right, start contemplating a suspension upgrade..