That is a nasty experience (for both you and the possum) but if you had swerved you might have come off very badly. Try not to feel bad.

The other thing with animals is, if they see you in time, they may swerve to avoid you. If you swerve the same way they don't have much hope. I was driving along a country road when I caught a glimpse in my peripheral vision of a small kangaroo coming straight for me. I braked and held my course, and he fortunately barely bumped the front of the car as he swerved away from me. He hopped off a bit shaken but no more than bruised.
At the same time, I probably saved my young daughter a severe injury and myself a badly damaged car by swerving when car came out at me from a left hand road many years ago. A young inexperienced driver, she was coming down a steep hill fast and her brakes failed to stop her on the gravel. I came a whisker within tipping myself upside down over an embankment but managed to get out of the swerve. I didn't stop - I was very shaken, and I figured if I got out of the car I might have been extremely unpleasant to this other young driver, and she looked pretty horror stricken at what she had nearly done.

I've been lucky twice in the last couple of months. Both times I managed to straddle a big carpet snake and a quail without harming them. Our road is narrow, hilly and windy so I frequently come over a hill or round a bend to find something on the road. With the carpet snake, he was curled up in the middle of the road in the pouring rain. I don't know how I missed him.

I was disgusted last year when we were coming home from a family outing and following another car, when the other car actually swerved onto the wrong side of the road to deliberately run over a large carpet snake. I wanted to get his number plate and report him but he was too far ahead to see.

I won't swerve to avoid hitting something either but the temptation is strong.