[COLOR="Blue"]JuneK33[/COLOR]

Yes it should be on the inlet side (inside the brass connection)
If not replace the valve as the screen is there to keep crud out of the valve which could hold it open and cause a flood.
Part number: AP2039339

I see the screen in the new valve. It can't be seen with the brass elbow in place. In parts of the cycle I get plenty of water, so I don't think I have a screen problem. It sounds like pavansut & I may have a similar issue with the machine draining at inappropriate times.
You mentioned above that the drain solenoid goes down for about 10 seconds and then the water pressure on the flapper valve keeps the drain open. My solenoid stays down for quite awhile. I have 6 fills in my cycle, and for fills 2 & 4 (seem to be the wash fills), the drain solenoid stays down until after the fill, so there is no water in the tub for the wash cycles. The machine runs for about 30 minutes after fill #4 before the solenoid drops to drain.

It makes sense. We bit the bullet & decided to replace the unit. It worked great for 24 of its 25 years. I may take a look at it before the new unit comes, though. Thanks for your help.

Sounds like you may have found the problem.

I would remove the solenoid assembly and see if that is causing it to not close all the way.

[COLOR="Blue"]JuneK33[/COLOR]

[COLOR="Blue"]I checked on the solenoid during operation of the dishwasher, and I can manually move it up when it has been pulled down, but as soon as I let go, it goes back down, until it decides to let go, then it moves back up again.[/COLOR]
It should not do this unless it is receiving power. And if it is getting power it will pull down hard.
Also if it is powered it will take a lot of force to move it to the up position, if you can do it at all. Also as soon as you let go of it it will slam back down., these solenoids are pretty strong.
It sounds like the spring that pulls the flapper back to the wash (up) position is weak.
Here is a link which shows how the springs should be attached. Click on the pump picture on the right hand side to get mare views. Note that some of the early units only have one spring (the long one)
http://www.applianceaid.com/ge-dw-pumps.html

Even when it is not powered but is draining there should be a fair amount of resistance as it is held in the drain position by the force of the water from the pump.

You should be able to tell when the unit is empty by the sound.
It will growl when empty and the solenoid should then move to the wash position.

[COLOR="Blue"]I took a quick look at the timer mechanism, and didn't see anything obviously wrong there, but don't know which cams operate the drain or fill.[/COLOR]
Usually with these units if the timer is giving power to the solenoid for too long of a period, the solenoid will overheat and melt the plastic on it and jam it up.

I am at a bit of a loss here.
But lets give it one more shot, this will involve laying on the floor for quite a while.
Put some pressure (not excessive) on the white solenoid/flapper lever to hold the solenoid plunger in the up position (wash). Be careful that when the solenoid pulls down it will not pinch your finger anywhere. Then lay there through an entire cycle with the pressure applied.
The unit should
Fill
Wash/rinse
Pull the solenoid down hard
Drain
Sounds normal and then growls
Solenoid moves up (wash) when water has drained out.
Unit fills
and repeats this for the different cycles.

[COLOR="Blue"]I hope denman can help solve it.[/COLOR]
I also hope denman can help solve it but so far he is batting zero.