The dishwasher will fill with water only in the first cycle, after it drains it will not fill for any of the remaining cycles. Is this a problem with the water valve or something in the timer that’s not telling the valve to open anymore?
Here are your parts
Replacement parts for WHIRLPOOL DU8350XT2 | AppliancePartsPros.com
Here is the wiring diagram
https://www.servicematters.com/docs/wiring/Wiring%20Sheet%20-%203368833.pdf
Only way to tell if it is the timer or the valve itself is with a meter.
If the valve is getting 120 volts and is not opening then it needs replacing.
Also check the overfill switch if no voltage on the valve. Brown wire on the switch to white wire on the fill valve, this will check if the timer is putting out the voltage. If not the timer needs replacing
One thing to check would be your float.
Perhaps all it needs is a cleaning.
Could be it is gunked up and sticking in the up position after the first fill, then over time drops. Normally the float does not come into play as it is a safety device but you may have above average water pressure/flow so the unit is filling more than usual.