For the cavity, the only option that I'm aware of is to have what is in the cavity removed and then redone with new insulation (e.g. certified bonded bead).
You will need the cavity filled in any case if you go with external insulation.
Imo, internal insulation should be the last resort.
Bear in mind that some cavity walls are constructed where the inner leaf wall is a block course higher than the outer leaf wall resulting in that top block of the inner leaf being uninsulated. If this is the case then the insulation hasn't dropped per se but a thermal bridge exists at the top of your walls. If this is the case then insulation can be added to link the cavity insulation with the attic insulation (how to do this depends on a number of things)
On the damp / mould issue, you are correct that the cooler wall surface higher on the wall is where enough condensation is occurring regularly to allow mould to exist. Improving your ventilation efficiency will help enormously with this so this should be considered too.