I was on my way home and an unleashed dog was crossing the road and when I swerved to miss it, I went into a ditch, causing a car accident resulting in serious injuries. Do I have a case?

Probably not.  Unless you can locate the owner or owners of the dog that were unleashed and running wild across the road, there will be no one to pursue a recovery against.  If the dog was owned by someone and they had homeowners insurance, and there is a leash law on county or city books, you may be able to recover.