How would I know if the motor mounts on my Crown Victoria are loose, because sometime after the car is running I'm hearing a strange vibrating noise coming from the engine?

If it's your motor mounts, the vibration won't stop.

You'll need to get under the car and take a look. The engine mounts are pretty easy to see - they'll have cracks through them. The Left one will be cracked and the right one will be squished. Use a pry bar and a flashlight and wedge between the Trans pan and the Cross-member while looking at the trans mount. Give the trans a lift and you can see the mount separate. We've been changing a lot of motor and trans mounts on our fleet of Crown Vic Police Interceptors. Not an unusual problem. Yes it will cause a vibration at Idle and yes, it can and will change as the Power train shifts position.