2007 Corolla LE - Vibrates and Vibrates Worse When Fog Light is On

[Corolla rock street]

I have a 2007 toyota corolla le when I start cold at morning the car won't vibrate because rpm was high to warm engine. When ever I swtich gear to D or R car will vibrate. Vibrate get worse when fog light is on. When I turn off and on fog light switch I can hear car engine going soft and hard when on and off. The car has 70,101 mile on it. Car had timing belt, spark plug , oil change and new battery change already. Also when I turn on fog light the rpm drop little bit and vibrate hard when I turn it off rpm go back up and just not vibrate hard but still vibrate.