Wow a lot of suggestions for new pads for an OP who said he has tried kool stop pads. My wifes entry level Trek bike also has horrible brakes, squeezing them they feel like mush, and afterwards the springs aren't strong enough to come immediately back to the starting postition. So she has horrible brake rub for 30 seconds after the initial braking. I adjust her brakes the same way I adjust mine, so I don't think it's an adjustment issue.
To the OP: almost any road caliper brakes will work fine with your levers. I would think a used set of 105's or Rivals off of ebay would be your best bet, assuming you don't want to spend too much money.
Joe