"develop white fur looking mold"
It is a salt crystal efflourescence caused by moisture seeping through concrete, leaching lime and crystalizing. It is removed with mild acid (vinegar, acetic).
Is this a drainage problem, and what is the solution?
Check gutters and downspouts around the garage to minimize water that can run under the slab. Be sure ground slopes away. Chances are you can improve the moisture situation through improved drainage. Hard to judge from where I sit.