I would imagine that this depends on your definition of "sin." There is the traditional definition that would have it be a violation of a rule of your Church or God ... from this viewpoint, his sin(s) is(are) obvious. In Chapter 17, the forest scene, it becomes clear to me that his sin doesn't involve what HE did, but what he did to Hester and Pearl --- he created a child and a loving relationship and then abandoned them both for his public image/career. No matter what he thought he deserved for his failing(s), Hester did not deserve the life of lonliness and alienation that was hers.
The argument that he could do more good if he remained a minister (and covered up his action(s)) is a bit convincing (it works in "The Minister's Black Veil" too), but, in the long run, seems to me to carry no weight. He should have done the right thing and run to Hester and Pearl (as Pearl requests over and over). Had he done that, in my reading, he would have been redeemed. And, of course, there would have been no book.
What an interesting question. Well, first off, he had a relationship with a married woman, engaging in an adulterous relationship. Even if Chillingworth had in fact been dead, he also had sexual relations outside of wedlock. He lied to everyone about that relationship. He denied his own daughter.
However, the thoughts that Dimmesdale has when he returns to town after meeting Hester in the forest suggest that there is much deeper sin going on. Dimmesdale is dishonest to all the town about how he feels. He passes judgement on them and has some desire to corrupt them, all of which he keeps to himself. Dimmesdale, in all areas of his life, is untrue to himself - which is probably the biggest sin of all. Here is the quote that expresses this:
Before Mr. Dimmesdale reached home, his inner man gave him other evidences of a revolution in the sphere of thought and feeling. In truth, nothing short of a total change of dynasty and moral code, in that interior kingdom, was adequate to account for the impulses now communicated to the unfortunate and startled minister.