Q: When engine is on in either reverse or drive the car bounces slightly but not in neutral or park and not all the time.

The front end is bouncy/wobbly when engine is on and car is in drive or reverse. There is no smoke, smell, noise only the uncontrollable movement. The movement is not violent but slight up, down, back and forth. I drive the car every day and it occurs at least once each day sometimes during the entire time the car is on and other times only for a short period. It can happen as I back out of the drive way after the car has just been started and has sat all night or not until it is driven home after sitting in the parking lot all day, each day varies. Can excessive heat have an impact as it is in the Arizona heat of 111 to 118 degrees when the issue started. I did a visual under the hood and under the car, nothing abnormal visibly and tire wear is normal. Pushed on the front end hood left and right above tires, had a stiff 1 bounce back. The same test on the rear had more spring with 1 bounce back. When in park /neutral the movement is lessened but still evident.

My car has 171000 miles.
My car has an automatic transmission.