OK. The ohm reads 241 on the red side and 284 on the blue side. So this means the water valve I ordered isn't needed. Right?
Yeah, chances are the valve is ok. It checks ok with a meter.
if it is just sitting there and not trying to harvesting the ice in the mold I am suspecting you have a defective module head on the ice maker, if this is out you will not get any water either since the ice maker has to cycle for it to give power to the fill valve to open and put the water into the icemaker mold....
Could not have said it better myself.
Do I need a new icemaker?
That's the way I would go. You could try to replace just the module on the end of the ice maker, but for the trouble and cost difference I would just replace the ice maker unit itself. Seems like once you have trouble with the module other parts (heater, mold, etc) are soon to give trouble too.
I have had the best luck with replacing the ice maker as a kit.
Here is the ice maker kit for your refrigerator.