Quick fix.
You say you cannot leave windows open all the time so you need to install trickle vents in the upvc window heads, it is possible to do this retrospectively but you have to know what your doing. If you cannot do this, physically fit air bricks in all rooms. Ideally you will have to fit two, one just above floor level and one just below ceiling in each room.
The heating will have to be on at a low temperature 24hrs a day.
You also confirm it is a solid brick house and not cavity. This gives me an idea of its age and type. You will certainly be getting cold bridging from the outside to inside. This would not have been a problem when the house was built as the open fires, which I presume you had which are now blocked up, would have created there own ventilation by drawing in fresh air through the gaps in the original draughty windows
and up through the gaps in floor boards and skirtings and drawing it up the chimney taking any moisture out with it.
Talking of which, As I suspect you have suspended wooden ground floors, please check your air bricks are well clear of soil or undergrowth, because with such a damp property I would worry that bad air circulation below the floor will allow dry rot to establish.
Also with a solid walled property it is imperative not to have ivy, or shrubs against the wall as the house must breath.
Long term.
It would be great if you could fit 25mm kingspan or similar insulation board the inside of the outer walls and then plasterboard over it for decoration, and that would save on heating cost and increase room temperatures, but that is a major construction job, however you will still have to have air bricks or trickle vents.
And finally, in answer to your question, Will this pack the house too tight and increase humidity?
I can see why you ask that, and my answer is NO. Because, insulation in itself will not increase humidity as it does not produce moisture, you and your family do that. But it will reduce the amount of energy you need to heat the house and it will stay warmer longer and more importantly it will remove another cold surfaces where mould creating condensation settles . The extra insulation will allow you to increase air changes by controlled ventilation, ie air bricks, trickle vents, extract fans etc without feeling the cold or horrid drafts.
Good luck.