My house was built in 1926 and others of the same design all had major problems when cavity wall insulation was installed. (I believe there is now a new method that works for these houses. )
So there is potential for problems with cavity wall insulation. Especially if it was done by a bit of a cowboy. The causes of damp is a complex subject It may pay you to consult an expert. There might be a simple and relatively cheap structural fix such as blocked air bricks or the damp proof membrane bridged. It may save you the cost of buying a dehumidifier and its ongoing running costs. I got damp in my single storey kitchen solely due to a hairline crack in a roof tile. It cost pennies to fix.
Even if the necessary remedial work is quite expensive it might still be better option to stop the damp at source rather than be perpetually fighting it.
How old is the house? It might not even have cavity walls. Buyers dont always get told the truth. It may need an expert to decide if the damp is rising from the ground, or running down from above perhaps due to faulty roof, gutters etc . Damp creeps and its source is not always obvious.
And if there is paperwork for the work being carried out, could it still be covered by a guarantee? 10 years ,maybe longer.