Those are my two best guesses.

The first - obligation - is simply that he was VERY close to Barton Cottage, had been earnestly and sincerely invited to visit, and could not think of a single good excuse not to. If dear sweet Mrs. Dashwood were ever to hear of his being so near to her as Plymouth and NOT coming by, she would be hurt. Edward does not wish to hurt anyone. And we can say that it is unlikely that she would ever hear, but we know how difficult it is to keep the action and designs of one part of the world from the notice and curiousity of the other ;-) I don't know if any of the Barton Cottage ladies is obliged to correspond with Fanny or John, but still people do tend to hear the most inconvenient things.