I live in an area with an abundance of deer. I've also spent 15 years driving up a canyon to my office and have seen too many crashes caused by deer, moose.. etc.
Being cautious has nothing to do with it. Most accidents are caused when the animal jumps in front of (or into the side of) the car. By swerving to avoid hitting a dear you run the risk of killing everybody in your car AND the other cars around you.
When a deer/moose/cow jump in front of your car you don't have time to check who's in the lanes next to you. I've driven past too many accidents that involved multiple cars because a driver swerved to avoid something that jumped in front of them.
It's also common for the animal to run into the side of the car while its moving... would you get a ticket for that as well?
I've taught my kids to avoid what you can... but when in doubt, don't swerve to hit the animal.