I have never had a Morgan rear axle to bits, but from the GOMOG page it seems that TAPER bearings are in use, and that an appropriately sized SHIM may be that which creates the required preload on the bearing to remove any play.
I suspect there will be a selection of different thickness shims available and a specified torque wrench setting after which there would ideally be no play felt in the assembly..?
IF the bearing is worn, fitting an oversized shim to tighten things up would cause the bearing to overheat and fail... !!!
1/2 inch movement felt at the top and bottom of the wheel would be worthy of inspection, however with knock on wheels I guess there is always a chance the wheel to hub fit, might be playing a part in creating the play felt....?
If there is oil escaping the axle, it might be worth checking that the breather is not blocked.
For sure if it has a Limited Slip Diff as shown in the GOMOG drawing, then LSD oil should be used
Hope as John typed above, someone with real world experience of the axle assembly on your model Morgan can jump in with better info to assist you Rene.
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