Coolant temperature sensor?

The green stuff is not good, what does the inside of the connector look like? Look really close into the female pins with a flashlight. See if the corrosion got down into the wiring harness. That might be your whole problem. If there's green crusty on the sensor pins then 90% of the time there is down in the harness connector too.

Look in the back on the connector too, the green corrosion works its way down into the wiring under the insulation and keeps growing down the wire. You can't even see it most of the time. Thats why we do so many different types of wire testing.

It depends on how far down inside the wiring insolation the corrosion goes. You can get terminal spray cleaner at an auto parts store and try that. But the Dark Green wire is only the ground wire for the Engine Temperature light. It will come on when the temperature gets too high to warn of overheating. I have that part of the sensor in orange below. The green/white wire is for the Gauge. As you can see it's a 2-part sensor. The other section goes to the engine computer to calculate the air/fuel mixture. With a multimeter you could test pin B to the housing of the sensor and see if there was any resistance reading on the Ohms setting. If the meter reads OL then the sensor has an open circuit inside. If there is a resistance reading, then there might be an issue with the wiring to the sensor going to the Instrument Cluster.
The corrosion causes high resistance in the circuit, and since this sensor works specifically on the resistance changing due to heat, this is why bad connections and corrosion cause big problems with this type of circuit.

Okay. I was thinking, since you're having so much trouble finding this sensor, you could try a vehicle used parts yard, if you look around, I'm sure you could find one. I know in my area there are a bunch with wrecked cars, and you can just go take the sensor out yourself and they only charge a few dollars. It might not be new, but it won't be broken.

That might be a safety feature. The engine computer will sometimes turn the fans on when it doesn't know what the engine temperature is. This way it prevents overheating. That tells you that the coolant temperature part of the sensor that goes to the ECM is most likely working okay. Is you check engine light on?

Most likely. The motor mounts are mostly made of hard rubber. But they wear out like everything else. You should be able to see movement in the upper motor mounts if they are visible.
Are you hearing a gurgling inside the vehicle while driving? That makes you think theres air in the system? An easy way to get air out is in the morning when the vehicle is cold, open the radiator cap, top off the radiator, and leave it open, then start the car. It will bubble any extra air out. On these older vehicles, I used to leave the radiator cap off until the thermostat opened. You could tell because the coolant will drop way down. Then I would top it off again and close it up. Just do not open it if the vehicle has been run that day. Only do this when its stone cold.

It's probably full then on coolant. Try turning the heat on too, if there is a diverter valve that is used for the heater core inside the vehicle, turning the heat on high will allow coolant to flow into the heater core inside the dashboard. I'll look up where all the motor mounts are if I can find them.

Yeah, I see a mount on the front of the engine, one on the rear, and there might be one underneath supporting the transmission.
Is the engine shaking a lot while it's just sitting and idling? With the hood open? Cylinder misfires will make an engine shake somewhat.

It sounds like there are some cylinders that are misfiring. When you shift it into Drive, you're putting a load on the engine and that's when most misfires show themselves. If it smooths out as you start to drive and have your foot on the pedal and you're opening the throttle plates it might be a vacuum leak. But when it's in Neutral or Park, there's no load on the engine. Granted the motor mounts may be worn down. I check those when a vehicle is turned off. Using a pry bar I'll go around and put some pressure where the mounts are and see if the engine will move more than it should. Using a long pry bar gives you some leverage to really check for play and worn-out mounts.

https://www.2carpros.com/articles/engine-misfires-or-runs-rough

https://www.2carpros.com/articles/how-a-motor-mount-works

https://www.2carpros.com/articles/replace-motor-mounts