Answer: He suffers in secret (hurts himself, refuses to eat, stays awake) and forgives himself of his guilt after the incident on the scaffolding where he holds a vigil where Hestor first suffered.
Step-by-step explanation:
I have yet to read the scarlet letter but i checked out some websites and i believe that Dimmesdale tried to atone for his sins by suffering in silence till the letter "A" appeared in the sky after the shaffolding vigil. Does that seem to be remotely close to what you're looking for?