On our Kenmore Elite Washer water continues to run into the tub during the spin cycle. I can cause the tub to drain and the spin to work properly by closing the water supply. If I turn the water supply back on, water begins to run into the tub during the spin cycle. If I leave the water supply on, the tub will never drain. Since the tub does not overflow with the water supply left on, the water level control switch must be working.

Look at these: https://dl.dropboxusercontent.com/u/...sher_Elite.pdf
https://dl.dropboxusercontent.com/u/...ng_Machine.pdf
https://dl.dropboxusercontent.com/u/...c%20washer.pdf
You will need to test some things. You need to know whether the water valves are getting power during the spin cycle. When it spins during the rinse water should go in, but not on final spin. You have a short somewhere. It could be where the wires are rubbing against the casing.