A Bone Seal Engraved with the Name Shaul, from the Time of the First Temple, was Found in the IAA Excavations in the Walls Around Jerusalem National Park, in the City of David. (complete article)
King Saul wasn't the only person with that name, so we can't say that the artifact was his.
Back to our first king...
To fully understand who King Saul was and why he failed, it's best to read the Bible. Read it as a narrative, without the commentaries. Start here in Samuel, and read it until Saul's death.
Shmuel Hanavi, Samuel the Prophet, was the leader of the Jewish People, but the people wanted a change. They wanted a king, like the other nations. they didn't ask for a king to observe the Mitzvah, the commandant of establishing a kingship, so Samuel objected. But G-d told him to listen to the people.
And then G-d "arranged things" so that Saul would meet Samuel and Samuel could anoint him king. From the text, it seems that Samuel was in charge, relaying to Saul the word of G-d. Whenever Saul took the initiative, he fouled up.
When things got very bad, G-d told Samuel to find another king. Samuel was told to get one of Yishai's sons, and that was David our second king.
This time Samuel let go and allowed David the chance to handle things on his own, and that included dealing with the then paranoid King Saul who was out to kill him.
David didn't have the impressive looks of Saul, but he had natural charisma and leadership.
We need a leader like King David. We need a leader who isn't constantly asking for advice and approval.