The best way to determine if the excess water is due to a water valve not closing completely, or if the ice maker motor is running slow, causing the valve to be open longer. turn the ice maker off for a few hours, just in case it just recently cycled, to allow the water to freeze. Now turn the ice maker on, and while watching it cycle, wait for the period it calls for water. There will be a cut out notch on the white main gear of the ice maker, the water valve circuit closes to call for water when the notch is at the 3:00 position. You will hear a faint click as it closes the valve circuit, allowing water to flow. the water should flow for a period of 8-10 seconds. You should once again hear a faint click to shut the water off. Ice makers normally call for water near the very end of the cycle, once the preceding cubes have been ejected. Whirlpool ice makers call for water when the ejector comb is at roughly a 1:00 position. If the water continues to run after this faint click, you likely have a bad valve. If you don't hear this faint click 8-10 seconds after the first, and its closer to 15 seconds or more, you have a bad ice maker.
It might help to turn the lights in the kitchen off, and to tape the freezer door light switch closed, as you can also watch the white center gear rotate, you will see a dim "spark" behind the white wheel when it calls for and shuts off the water, both previously mentioned faint clicks.

telling me that makes me believe in more so that your ice maker motor is slow, and that is the cause of the problem.