Why does my filtered water turn thick and almost slimy when refrigerated?
Your setup does not effectively filter nutrients. Nutrients can be problematic in climates like Charleston, SC which are humid.  Humid climates plus a nutrient content in the water create the potential for fungal growth. 

Part of living in humid climates is dealing with fungal growth. Fungal growth starts from spores (capable of growing in water) which are probably living in the refrigerator.  Disinfecting the refrigerator should be done by the instructions in the manual which came with the refrigerator. 

This should go without saying, the water with the starchy taste should probably be thrown out. Any drinking water left to sit has the potential to grow micro-organisms, so storing water in a refrigerator is a bad idea. Try making ice cubes in the freezer and adding it to water if you want cold water.  

In humid climates, anything stored in the refrigerator should be sealed so spores cannot grow.  The refrigerator must be cleaned regularly. 

Filtering water on an as-need basis may be best with your setup because a carbon filter is effective at removing anything biological.  Personally, I use a pitcher because I can refill it as needed, let the whole setup dry periodically, and easily throw out saturated cartridges.