Did Saul's family die with him?

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Yes, all of Saul's family died with him.

So Saul died, and his three sons, and all his house died together. 1 Chronicles 10:6

No, one of Saul's sons (Ishbosheth) became king of Israel after Saul died.

Your master Saul is dead .... But Abner ... took Ishbosheth the son of Saul, and brought him over to Mahanaim and made him king. 2 Samuel 2:1-7

No, David sacrificed two of Saul's sons and five of his grandsons to God to stop him from starving people to death.

Then there was a famine in the days of David three years, year after year; and David enquired of the LORD. And the LORD answered, It is for Saul, and for his bloody house, because he slew the Gibeonites. ...Wherefore David said unto the Gibeonites, What shall I do for you? and wherewith shall I make the atonement? ... And they answered the king ... Let seven men of his sons be delivered unto us, and we will hang them up unto the LORD ... And the king said, I will give them. ... The king took the two sons of Rizpah the daughter of Aiah, whom she bare unto Saul, Armoni and Mephibosheth; and the five sons of Michal the daughter of Saul ... And he delivered them into the hands of the Gibeonites, and they hanged them in the hill before the LORD: and they fell all seven together, and were put to death. ... And after that God was intreated for the land. 2 Samuel 21:1-14