Hi JGMC, Looking at your pictures it looks like there is an extensive distribution of mold. You are right, to eliminate mold you have to increase air movement and decrease moisture.
One eco-friendly option would be to install a power free roof ventilator.
Here is a good visual aid of how roof ventilators work at increasing the air movement in your house.
But these will not be effective for the underground level of the carport as a powered air ventilator will be needed.
A good eco-friendly option is solar powered ventilation.
Assuming you mainly use the carport for storage, temperature control will not be important, but only ventilation.
Additionally, it would be good to install a humidistat so that you can get reliable humidity levels in the room. This way you will be able to adjust the ventilation accordingly, and have an objective way of measuring the effect of the ventilation.

I forgot to mention the garage is on the lower level and the house sits on top of the garage, which is made of concrete and steel beams.

Impossible for me to comment without actually visiting the site--it seems pretty complicated but I don't think you have many options other than to find a way to circulate air in there. You might need an exhaust fan as part of the solution but since you don't mention standing ground water I don't think you need a perimeter drainage system. You should get a water-proofing specialist to come and give you a consult.