What was King Saul's full name?

Quite likely "Saul" was his full name. Unlike today, a person in that day & area would be known by a single name, perhaps with a reference to his home town or his genealogy. For example, the Apostle Paul was originally known commonly as "Saul of Tarsus," meaning his home was in Tarsus. King David might be called "David, son of Jesse."