Hoping I can find some help on here!
I seem to be having the same problem an individual listed here: http://forum.appliancepartspros...r-during-spin-cycle.html , but with a different washer. I have a Whirlpool Ultimate Care II (8 cycle, 5 speed. Can't find model number.). Our washer agitates the second we turn it on, without filling with water first, and then, during the spin cycle, it continues to let water in. The water is draining out, but still leaves the clothes wet.
Normally I wouldn't let the problem bother me, as we bought our washer second hand, and I can simply spin the load again after turning the water off, but my husband and I cloth diaper, which means that we have hard water soaking into our diapers, which results in absorbency issues. (Oh the drama.)
Any advice on parts or how to fix the problem would be much appreciated!

Is this a washer that is supposed to sense water level? Meaning it doesn't have a water level switch or load size switch with low, med, high etc...? It would have a soil level switch instead.

Looks something like this image?

Another way to ID it would be if normal behavior is to fill partially, agitate for a short period and then fill to whatever load size it senses?

If so it's the sensing switch, they fail in exactly that way. If you slip a putty knife or other slim object under the front bottom corners of the console and gently press towards the back of the machine to release the console clips the whole thing flips upward. Your bad part would be the square black switch with a clear hose going to it and 5 or more wires. There's 2 versions of that switch. I think it has a part # on it. Either W10248240 or w10292584 I think. One is larger than the other. They both fail at an amazing rate.

If it has a water level or load size switch and a round white part with a hose and a few wires inside it is a standard pressure switch and you can ignore what I said.