First, I don't know what your budget will allow. Second, the main way to remove moisture from air is by having a surface that is cold enough to condense water out of the air.
Other than the technology for cooling something you won't need air conditioning. You should have an aircirculation system that actively moves air throughout the house. At this point you can't do anything about it but what was used and how was moisture proofing applied to the outside of your house? It might be interesting to know the source of all that humidity.
How much outside air are you bringing into the house? Outside, summer air alswys has moisture content that makes higher humidity when you cool it a few degrees. You won't be able to have open windows or more than a basic level of fresh air ventilation. In other words your moisture is coming through the walls,floors and ceiling from the ground or it is coming into the house in the incoming fresh air. On top of this you must have a minimum level of ventilation to stay ahead of indoor air pollution and radon accumulation. It shouldn't be difficult to process and dry the incoming air if it is coming in through one intake. Then the air distribution system can deliver the dried fresh air throughout the house.
The thing that will bother is that the most effective way to dry the fresh air takes electric powered refrigeration. A dehumidifier works the same as full AC to dry the air. A dehumidifier basically cools the air to remove the water and then warms the air with the refrigeration generated heat. In the cool season when the incoming air is warmed the drying happens at no cost.
One way to reduce the operating cost of running the dehumidifier is to use cool groundwater to carry away the surplus heat. This lets the cooling part of the system work easier and with less expense.
It might work out that the dried incoming air can pick up moisture coming through the shell of the house and carry it either away or to the dehumidifier to be removed. Since fresh air ventilation through a system is a must in your house, I'd start the dehumidification in the freah air ventilation system.
JdN
