Maybe implement a counter of how many consecutive preceding rounds all players have been counselors. The player who has played the most consecutive preceding rounds as a counselor is determined to become Jason, and their counter resets to zero. When there's a tie, randomly choose one of the tying players. This doesn't address the ragequitting aspect, but it might be a good way to start, though this has probably already been thought through.
In the films, Jason always had a knack for catching up with people unreasonably quickly, even though he is (almost) never seen actually running. For example, the mast-climbing kill in part VIII, or the scooter double-kill in part VI. Perhaps Jason's movement speed should increase when he's out of a direct line of visibility with the counselors. Just a thought...