Here is your parts breakdown, includes a wiring diagram
Model LAT8234AAE

I would try different temp settings in the wash mode, hot only and cold only.

Could be one of them is not operating correctly. In warm both valves will be open so you will get lots of water even if one is plugged.

If one setting does not give good flow
Remove the hose from the tap and run it into a pail just to check the water supply.
IF OK, remove the hose from the machine, check that it does not have a screen filter built into one ends rubber washer.
If OK check the filter screen on the inlet side of the machine's water valve. Clean it if it is clogged up. Do not remove this screen, it keeps crud out of the valve which could hold the valve open causing a flood. Also if it is the hot side that is clogged drain some water out of your hot water tank use the drain or the pressure release. Let it run till it is clear.
If OK reattach the hose and run this into a pail, sometimes a hose can collapse internally.

If all the above are OK then odds are high that a water valve is not opening fully.

Here is a service manual
http://www.servicematters.com/m...6010373%20(16001145).pdf

Take a look at 5-26 it has the wiring diagram and a timing chart.
If you look at the timing chart you will see that the rinse fill is a separate set of timer contacts.
Odds are high that these contacts are the problem and the timer needs replacing especially since you also see an arc mark on the timer.

If you have a meter you could confirm this by:
Unplugging the machine
Setting the timer to where it should be filling on rinse
Unplug the connector to the timer
Measure across the contacts. Should be 0 ohms.

Would also visually check the timer connector.