How can one get rid of recurring doubts about the existence of God and the truthfulness of the Message, despite trying to repel them by keeping busy or reminding oneself of the evidence?

These whisperings do not harm you as long as you dislike them; rather, you will be rewarded for resisting them. They are from the stratagems of Satan, so you must strive to repel them. If Satan casts any of them into your heart, cease thinking about them, and say: "I believe in Allah," and seek refuge in Allah from the accursed Satan. Turn away from these whisperings and think about what benefits you in this world and the Hereafter, and resort to Allah Almighty to remove them from you.