Was David responsible for Saul's insane order to slaughter everyone in the city of Nob (1 Sam. 22:22)?
No, not really, but the buck stopped at David.
1 Samuel 22:
20 But one son of Ahimelek son of Ahitub, named Abiathar, escaped and fled to join David. 21He told David that Saul had killed the priests of the Lord. 22Then David said to Abiathar, “That day, when Doeg the Edomite was there, I knew he would be sure to tell Saul. I am responsible for the death of your whole family.
Abiathar was in tremendous distress. David personally apologized to him and accepted responsibility. David was not admitting this in front of Saul or Doeg. He was speaking comforting words to Abiathar.
23 Stay with me; don’t be afraid. The man who wants to kill you is trying to kill me too. You will be safe with me.”
This contrasted Saul's behavior 7 chapters earlier in 1 Samuel 15:
7Then Saul attacked the Amalekites all the way from Havilah to Shur, near the eastern border of Egypt. 8He took Agag king of the Amalekites alive, and all his people he totally destroyed with the sword. 9But Saul and the army spared Agag and the best of the sheep and cattle, the fat calves and lambs—everything that was good. These they were unwilling to destroy completely, but everything that was despised and weak they totally destroyed.
Saul sinned for failing to completely followed God's command.
13 When Samuel reached him, Saul said, “The Lord bless you! I have carried out the Lord’s instructions.”
14 But Samuel said, “What then is this bleating of sheep in my ears? What is this lowing of cattle that I hear?”
15 Saul answered, “The soldiers brought them from the Amalekites; they spared the best of the sheep and cattle to sacrifice to the Lord your God, but we totally destroyed the rest.”
Excuses, excuses. Saul was the king; he could have ordered the soldiers to follow orders. Instead of taking personal responsibility for his own sin, he blamed his soldiers.
Was David really responsible for the acts of a lunatic like King Saul?
No. Yet, David accepted responsibility contrasting Saul's shifting responsibility to his inferiors.
Did the blame not fall squarely on Saul and his executioner, Doeg the Edomite (1 Samuel 22:22)?
I think so and many others also in 1 Samuel 22:
17Then the king ordered the guards at his side: “Turn and kill the priests of the Lord, because they too have sided with David. They knew he was fleeing, yet they did not tell me.”
But the king’s officials were unwilling to raise a hand to strike the priests of the Lord.
They disagreed with Saul.
18The king then ordered Doeg, “You turn and strike down the priests.” So Doeg the Edomite turned and struck them down. That day he killed eighty-five men who wore the linen ephod. 19He also put to the sword Nob, the town of the priests, with its men and women, its children and infants, and its cattle, donkeys and sheep.
Those who disagreed with Saul didn't think that David was responsible but Doeg was responsible for the killing.