Season 8 "A Girl For Goober" - Though not the last episode to air, it was the final episode filmed in the series. According to "Andy and Don," published in 2015, the episode was recorded on February 21, 1968, and was followed by a huge wrap party at Andy Griffith's Golf Club in Toluca Lake. (page 181, "Andy and Don," Copyright 2015 Simon & Schuster).
This was the last episode that Ron Howard appeared in as Opie Taylor. He wasn't in the final episode.
Goober claims to be 33 in this episide. In an earlier episode called Goober, The Executive he was 36.
Dr. Gibson mentions the towns of 'Toast' and 'Manteo.' These are actual towns in North Carolina, and are not fictional, like Mayberry.