What are you drilling into? Sounds like something like oak, or else your battery (if it's a cordless drill) is underpowered or needs recharging.
The problem isn't the screw material, the problem is the method you're using, or the tool itself. Stainless is a waste of money for a cab, and even those zinc ones you're using are higher-priced than need be. Standard drywall screws are just fine, and generally cheaper than the zinc ones.
Where'd you get those from, Menard's?