How did King David overcome his errors?

Answer 1

King David overcame his errors by admitting to them and taking responsibility for them. When Saul committed an error, he blamed it on the people. When he took the Amalekite cattle as booty when he was told not to , he blamed it on his soldiers and refused to take responsibility.

In contrast, when David was confronted by the prophet Nathan about his sin of taking Batsheva and sending her husband Uriah to the front after David had impregnated her, David immediately admits to his mistake and says "I sinned." This is how he overcame his errors and why God established the Davidic dynasty through him.