What make and model?

Yes, there is a sensor. But we need more info

It doesn't sound like a sensor issue to me - if it were, the washer would sit around doing nothing, waiting to be informed that it is full; you mentioned that it "goes into the next cycle", so it's obviously aware that it's full. It just can't shut the water because of a physical restriction - most likely some debris is clogging one or both of the solenoids (the electronically-operated valves that a washer uses to turn the water on and off), jamming it/them in the "open" position even when the machine wants it/them closed.

Here's my suggestion: re-post this question the Appliances page; we have some top-notch appliance repair guys there (appleguy in particular comes to mind) who can probably give you more specific advice. Make sure to post the manufacturer and model name/number of your washer, since solutions to such problems often differ depending on the design.

Good luck,
Moishe
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This question has been reposted over in Appliances. Tom