I am assuming that the pump does run during drain and that the wash cycle runs normally.

[COLOR="Blue"]checked that the drain hose is clear,[/COLOR]
Remove the drain line from the house plumbing, pull it back through the cabinets and then drain it into a pot/bowl, gravity should empty out most of the water. If not you have a plug in the drain system.
Did you check it all the way back to the sump?
Remove it and it's fitting to the sump and check it.
Have you checked the house drain/ dishwasher hook up fitting.
One way to be sure this is not the problem is to undo the drain hose at the house plumbing and try a drain cycle into a pail. Be sure to hold the hose securely in the pail while doing this, also put a towel over the pail to stop the water from splashing all over, unless of coarse you want a shower.

[COLOR="Blue"]watched the solenoid action,[/COLOR]
I am assuming the solenoid plunger is pulled down and stays down while it tries to drain.
Is the water still being directed up to the wash arms?
If yes it could be that the flapper valve in the pump assembly has come loose from the actuation arm.

[COLOR="Blue"]cleaned the basket[/COLOR]
I am assuming you mean the sump in the bottom center of the unit.

I would empty the unit of water so you can properly check the drain components. You should be able either undo the drain hose and empty it into a pot/bowl or siphon out most of the water. A turkey baster comes in handy to get the last bit of water out of the sump.

Here are your parts
Replacement parts for HDA3600 models | AppliancePartsPros.com

Just to double check. I went through the parts list and it does not look like this unit has a Piston & Nut assembly but wanted to double check since I could not find a parts breakdown.
Below is a link to a unit that has this Item 564 in Section 2. This is often a cause of poor draining.
Body parts replacement parts for General Electric GSD5134D00WW Dishwasher | AppliancePartsPros.com

If you click on the picture of the pump on the first page of the parts a new page will open with more views. If you look at the down the throat picture you will see that there is a black seal in the center of the flapper valve. Check to ensure this is there.

[COLOR="Blue"]Does this indicate a bad pump? The unit is pretty new, but maybe...
The motor seems to be rotating at a good clip, but might that be the problem?[/COLOR]
I think it pretty well has to be a pump problem as you have checked everything else. I would pull the pump and check it. Perhaps the impeller is slipping on the motor shaft or something similar.

[COLOR="Blue"] Is that the seal you are talking about? If so, it is definitely missing from the flapper in my pump. Is that the smoking gun?[/COLOR]
Yes, I think it is. With it missing water will be pumped back through the machine rather than out the drain.

[COLOR="Blue"](Also, my coupler - part 450 - is destroyed. I plan to replace that, but I don't think that will cause a draining problem - will it?)[/COLOR]
Cannot say for sure on this but anything in the water flow path that is messed up will contribute to your problem.

FYI: I am off to the lake for a few days so will be unavailable. Sometimes when this forum gets busy none of the regulars here have time to look at a post if they see another regular is answering so if you still need help and do not get a timely response, you may want to start a new post with a link back to this one.

The ball is VERY important to the drain function on your machine. It has to seal the bottom of the filter shaft to prevent the water from recirculating back through the filter.

It could be the motor but may also be something jammed in the impeller/chopper.