For this to occur, the water valve must be the problem. It contains not 2, but 4 different valves and solenoids to dispense the bleach and fabric softeners. The bleach solenoid section of the valve has evidently failed. This is not a cheap part, since it also contains a temperature thermister, but fairly easy to replace.
Since the dispenser cups on this model are gravity fed, two things must be observed:
1. The machine must be level
. If not, liquid will not fill up the dispenser reservoir properly so it can siphon off once it reaches an overflow tube under the cover. An unlevel machine will cause poor emptying of the dispensers, or in some cases, failure to siphon at all.
2. The cold water supply
coming into the washer must be strong in pressure. If the inlet hoses or screens are clogged, water will trickle into the washer too slowly to flush out the dispensers.
If the bleach thing is not too bothersome, I'd leave it alone. To me, that's not worth a $100 part.