Work out where all the bearings are in the axle:
a wheel bearing at each end, two big bearings one either side of the differential itself, and a double-bearing on the input shaft.
It sounds like one or more needs to be changed, and if it was me, from your description of what happens, I would suspect the bearing in the nose-piece of the diff is running a bit loose.
I say this because the noise seems to be load-sensitive, suggesting the pinion is able to move in relation to the crownwheel, which is a BAD IDEA.
If you are lucky-ish you MIGHT be able to improve things by tightening up the nut in the middle of the prop shaft flange on the diff. That nut should be pretty damn tight, so if it is loose ..... wonder why.
Tighten it anyway of course.
CharlesY