The aircraft was being repositioned for an air taxi operation. The pilot began taking off on a slush-covered runway. He reported that as the aircraft accelerated, the nosewheel began to sling water and slush off the runway and onto the windshield. He raised the nosewheel slightly to eliminate this distraction. At about that point, the aircraft swerved to the right. Reportedly, the pilot immediately moved both power levers back to the reverse to abort. The aircraft then swerved abruptly to the left, but continued off the right side of the runway and went down an embankment. The pilot brought both condition levers back to the emergency shut-off position as the plane skidded off the runway. Frozen slush was found on the airframe. A grayish-white residue, possibly from evaporated water/steam, was noted inside the exhaust stack of the right engine. Both prop feather valves were found open, both engine shut-off valves were closed, both prop latch pins were engaged, but the left prop was feathered and the right was in flat pitch.