Good effort Jen for your first efforts at panning
I would use a faster shutter speed to start with, say around 1/200 or 1/160 so you are getting a much higher rate of "keepers" with most of the car sharp. You can then move onto slower shutter speeds as the event progresses.
There will always be some parts of the car out of focus when panning, unless the car is moving completely parallel and level to you. I did read the reason for this somewhere, but I recall it's to do with the distance between you and say the front and rear of the car changing as the car moves towards or away from you. This effect is more pronounced the closer you are to the car. If you are using a long lens and are further away, the % difference between the distances from when the shutter opens to when it closes is much smaller. With slower shutter speeds, you cna often end up with just a very small part of the car in focus.
John
you'll never be happy, it's a disease