This will sound obvious, but you need to work out what's actually causing the damp before you try to fix it.
There are various things which might cause damp - the first is condensation (basically water in the air hitting a cold surface and condensing), then there is penetrating damp (i.e. leaks) and then good old rising damp, which is water coming up the walls, which is what your damp proof course is for. Some people would say vociferously that you are much, much more likely to have something other than rising damp, but you tend to be sold solutions to rising damp.
It sounds like you haven't had much ventilation in the house. I'd guess that the problems are condensation - you're putting a lot of water into the air through showers, baths, breathing, cooking, etc, and then this water can't escape because you can't open the windows and didn't have an extractor fan in the kitchen. When the moist air hits the cold walls, the water condenses out. If you can get more ventilation, you should see things improve, but you're going to have to give your walls time to dry out - I think the rule of thumb is that it takes a month of summer for an inch of damp wall to dry, and that's with it not getting much new water into it.
However, you really must go all over the house and attic checking for leaks, drips, broken gutters, etc. Best to do it when it's raining - kids actually quite enjoy this! Then check again at least once a year to make sure you don't get damp problems coming back
HTH