Surplus part came with the dishwasher

My best guess is that the impeller has worn down enough that it no longer makes a good enough seal to act as a check valve. That's drain water seeping back in.

The dryer on a DW is that metal rod running along the bottom. The hole on the side is the fill inlet.

If it is fresh water leaking in, having replaced the valve the only thing left is the control board. The original purchase price, replacement cost and/or repair parts availability and their cost will dictate your economical options (don't buy a $100 control board for a $200 DW).

Replacing the impeller is a relatively cheap lets see if this works solution; replacing the control board would be of the this had better work kind.

Tracking this down may be more work than it's worth. It's not uncommon for most dishwashers to have some water remaining in them. If it's not spilling onto the floor or growing mold, who cares? If this was a friend-house-call, I'd tell you to (a) call me back when it starts leaking on the floor (b) buy an impeller kit (c) stop throwing money at it and buy a new one (the cost difference between new dishwashers is noise level; they all work just fine).

That is of course, after having insured that the drain line is elevated to reduce back-flow and that it is free from blockage by doing a garden hose enema with the impeller removed. If you're lucky, there's a grease plug fouling up the works.

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The dishwasher does not know if you have a ground or not. The ground wiring on the inside of the machine (the dishwasher in this case) is tied to everything that will conduct electricity easily and could come into contact with the user; typically anything metal.

A ground wire can be thought of as simply a backup neutral. So if a hot wire ever touches some part of a metal system, the current will have this backup wire to run the current back to the breaker box. The breaker will detect this increase in current (unimpeded path) and trip which will cut off power to the device. If this wire weren't here, the metallic object would be energized, but it would not trip a breaker. It would simply be waiting for something (a person) to come along and touch it to give it a better path to ground.

So, as long as a hot never comes loose, you wouldn't need the ground. But if it ever were to, this wire could save a life (or prevent a really strong tingling sensation).

To shorten things off and end this; you don't need the ground for the dishwasher to function. It will work perfectly fine without one and you just wouldn't connect the ground wire on the other end. You then run the risk of a wire coming loose though.

But, what should be done regardless and will also provide grounding functionality is to use a GFCI breaker. The breaker would replace the existing breaker feeding the dishwasher and include a pigtail to the neutral. This breaker detects the current that is feeding into the circuit and out of the circuit. As long as everything is running like normal, it will work like a normal breaker. However, if a hot ever comes loose, the GFCI would detect that current is going in, but not as much is coming out and the breaker would trip itself.