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As far as the packing nut turning and not getting any tighter, it could be because the bushing the packing nut threads on to is loose in the pump housing. If so, you should be able to see both turning as you try to tighten the nut up.
If that's the case, you need to remove the pump and either get a new bushing or, if the bore in the housing is worn, another housing.
Also, just wondering if you've stuck a thermometer in the top of the tank (through the filler) to see just how hot you're getting. Could be that there's just a bit too much water in the radiator and the excess is getting pushed out the overflow. Either way, good luck and let us know how things turn out.