If I totaled my vehicle due to livestock in the road, is the owner of the livestock owner responsible for damages?

Your claim against the owner of the livestock would be negligence, if that owner had a duty to keep the livestock out of the public roadway, and by not doing so the livestock caused a public endangerment that caused you or your vehicle damage. Speak with an attorney in your area who can determine if you have a valid claim based on your state law.