Not only coincidence. Have you ever heard thinking of something bad will bring it to you? If your paranoid about being haunted, your mind will create the imaginary scene as if you are. Such a powerful thing we have.
Still, I've never heard cleansing can hurt anyone so why not cleansing? There's nothing wrong with it especially since its second hand, you can cleanse it from all the energies attached to it by the former owners.
But really, not every single thing you buy is cursed or haunted. Don't force your mind to build an illusion.
Cleanse the house and the car. After the cleansing, do something to attract the positive back into the home. Otherwise, it's just a blank space. Many people forget that step. Then protect the car and home spiritually.
Although I doubt the car was cursed or had anything to do with the illness, it still helps. It feeds the same thought processes that make you believe you are cursed.