When the apostles passed off the scene, did they give any kind of hint or clear instruction as to what believers and churches should look to for final authority in matters of faith and practice?

Yes, they did–note this comment by C. I. Scofield on 2 Timothy 3:16-17…

The apostolic presence is about to be withdrawn. That presence had carried with it authority, because the apostles were inspired and sent forth as the spokesmen of Christ. Looking forward now to his approaching removal by death, the apostle points the servants of the Lord to the Scriptures as inspired, authoritative, able to make wise unto salvation, and competent to teach, reprove, correct, instruct, and to thoroughly furnish the man of God unto every good work. Not one word does he say of a successor to his apostleship, nor of the authority of a church, body of churches, council, or creed. The Scriptures, and they alone, are authoritative (Scofield Correspondence Course: The New Testament, p. 305).