I have Honda accord 2013. 73000km on it. When I first turn the car on and the engine is cold, I can have the car stopped (in gear) and it will not shake. After the car has heated up, car begins to shake. If I turn the fan on, the shaking gets lower. In neutral or park the shaking diminishes.
Worn engine mounts? try this if equipped with auto tranny. open the hood. one person starts engine; engage the e-brake and left foot pressing hard on the brake pedal. Put the car in drive gear and right foot blips the gas pedal. The second person stands to the side of the open hood and watches if the engine jumps up and down.
I get nervous when I'm at a red light stop too. Car is probably feeling the same :).
On a serious note, I would start with cleaning the throttle body. Also check the condition of the air filter while you are at it. When the engine is cold, the idle is a bit high. As the car warms up, the idle speed lowers. This might explain why the car does not shake when cold.
Also when you put it in gear this engages the torque converter. This adds a slightly bit more strain on the engine and couple with the low idle causes the shaking.
I cannot explain why turning on the fan would help the situation. Does the car overheat?