Axel shaft hot

You wouldn't have an exhaust pipe aimed at the axle ?
Good question! That is such an obvious thing I never would have thought of suggesting to check the exhaust.

As have most others here have already mentioned, I immediately expect it to be a bad bearing. My only thought about that, is it takes very little roughness in a bearing to produce heat and noise under load. Thus it probably makes sense to change the bearing even though it may feel just fine in your hand.