The primary reason Saul continued to chase David throughout the land of Israel was due to his intense jealousy and fear. After David's military victories, especially the killing of Goliath, and the women’s song that celebrated David's accomplishments over Saul's, Saul became deeply envious. This jealousy was compounded by the realization that God had chosen David to be the future king, which threatened Saul's own kingship.
Saul's obsession with eliminating David led him to chase him relentlessly, even beyond the borders of Israel, as he believed that David was a direct threat to his rule. Therefore, the most accurate answer aligns with Saul being "half-crazed with jealousy towards David" and feeling that David's burgeoning popularity and favor with God and the people would undermine his own reign.