the damp marks look to me like dot and dab plasterboard, transmitting cold to the plaster by conduction from the cold wall through the blobs. Hopefully they are not tranmitting actual damp.
plaster dries out by ventilation not heat. turning heat on will just waste money. ventilation is cheaper than humidifiers. you can place a room fan or desk fan to blow air against the wall, this will speed up evaporation like hanging washing on the line of a windy day. ventilation will then take the water vapour away.
if this is an actual house, open the loft hatch and you will find natural convection drives an air current up through the house, into the loft, and out through the eaves.
water vapour is lighter than air (hence clouds) so it will naturally rise away if you let it.
it will be cold because as water evaporates, it absorbs heat.
when the house is occupied you can open the downstairs windows to encourage airflow, but usually old houses are quite draughty, gaps under the doors, letterboxes, airbricks in the kitchen etc so enough fresh air will enter. Open trickle vents on windows if there are any.
after a week you should see the plaster drying to a pale matt pink. shiny or chocolate patches are wet. Plaster is made of water, so needs drying out.
the windows will get misty or wet overnight until the drying out is finished
Only when it is dry is there any point in using mould killing chemicals and scrubbing clean. The entire world is full of mould spores and they will grow wherever they find warm damp conditions.