The substance from the moisture is fluffy yellowish white power and seems to grow.
That is efflorescence. Efflorescence is basically the rising of natural salts that occur in the cement products. It rises due to the presence of moisture. The moisture is naturally seeking to evaporate so it rises and carries the salts with it along its journey, and once it arrives at the surface it tends to blossom and bloom. If left unrestrained it can almost look like cauliflower in some cases. Anyway that's very basic explanation.
The existence of moisture can be totally natural and normal. You don't necessarily have to have a sub-grade water leak to be experiencing efflorescence but a water leak could also be the driving force causing the efflorescence.
Concrete is also a natural magnet for moisture so to speak. The depth of the water table can have an effect on the moisture in the concrete but so can the environment.
Based on your comments about a "wet winter" and your neighbors not being aware of the same issues, this points to an isolated problem with your property. More than likely your landscaping.
I think your insurance company will back away from this problem shortly and leave you to deal with it. The musty odor is likely from a mild mold/mildew condition. You are doing most everything you can at this point. Dehumidifying is probably going to be your best bet.
Calcium chloride testing will tell you how much moisture is in the slab at the time of testing but that doesn't mean that degree of moisture is there year round. Normally it won't be, it (moisture) can and does change with the seasons. Ca Cl testing may be a waste of money.
The house is fifty years old but how long have you been there. Out of curiosity...where are you in California? East or west of the hills?