What constitutes "not driving with due care"?

Question Details:

I was given a ticket for "not driving with due care" when my car went into a ditch and the flipped over. I was trying to avoid hitting a deer that jumped in front of my car. My insurance has a required charge especially for this problem because of all the accident caused by deer. I was driving the speed limit or below, there were no other cars or people involved and no property damage. Additionally, because the police detained me so long, I was unable to go after my little dog that had been in the car with me. She has been missing for almost 7 months now.

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