If anyone is thinking of getting their homes insulated you should bear in mind the following:
Always go with a reputible, well established company.
Make sure that a complete thorough survey is undertaken (this should include a boroscope check and combustion vent check)
Ask what material they use, It must be superior quality (Knauf Supafil mineral fibre will not allow damp or cavity bridging to occur under proper conditions)
A good company will also colour match the mortar in the drill holes.
And make sure they are backed by a 25 year C.I.G.A. guarantee. This ensures that should anything happen you are covered.

There are many benefits in getting your home insulated, including saving money on energy bills, more comfortable living environment, reducing co2, reducing carbon footprint, improving energy rating (EPC)
A company I would recommend is Mark Group – a national based company that does all of the above.

http://www.markgroup.co.uk

I am a cavity wall installer myself and if you have had your walls already insulated then you need to get in touch with the company that installed the insulation to investigate the problem.
Not being able to pump any more insulation into the wall is a good sign as it means that it has got enough in.
If you do not know the company that installed your insulation then you need to get in youch with CIGA and they should be able to help you.

Could you elaborate on what kind of damp you are experiencing, ie. is it a patch in a certain place, black mould, can you feel moisture on the inner wall. What room is it damp. Is it on a certain wall.

Most damp problems are due to cavity being bridged by rubble. When the insulation is pumped in, the insulation drops, hits the bridge in the cavity and causes a void where no insulation goes. This is what we call a cold spot and can cause damp due to the intensity of the heat from your house going for the cold spot and condensating.

Also just so you are aware. Dont let anybody try pump anything other than rockwool into your cavity if that already exists. You cannot mix insulations due to the differant U values and this can cause damp also.

To Sum Up… Contact company that installed in the first place. Or contact CIGA if you dont know who will help you as it should be registered with them as done and by who.