One of the questions that dominates "Babylon Revisited" is whether or not Charlie is reformed. You can make a good case either way.
On the one hand, Charlie is adamant about having only one drink per day. Even at the very end of the story, when he's lost his chance at getting Honoria back, he refuses to have a second cocktail. He sticks to his guns, and that suggests that he's serious about becoming a new man. His attitude toward his past behavior is definitely that of a reformed man. He is disgusted at and horrified by the way he used to act, as we see clearly in these passages:
All the catering to vice and waste was on an utterly childish scale. (1.56)
His first feeling was one of awe that he had actually, in his mature years, stolen a tricycle and pedaled Lorraine all over the Étoile between the small hours and dawn. In retrospect it was a nightmare. […] How many weeks or months of dissipation to arrive at that condition of utter irresponsibility? (4.13)
Also, Charlie is adamant in his refusal to go out partying with his old drinking buddies, Lorraine and Duncan. He wants nothing to do with them. As for his relationship with Honoria, we get the sense that Charlie would make a good father; his understanding of his daughter and the goodness of his intentions is never thrown into question.
On the other hand, the first thing Charlie does when he gets to Paris is go back to his old bar and start looking for his old drinking buddies. He even leaves the Peters' address for Duncan at the Ritz so he can get in touch later. While he only has one drink per day, Charlie is still drinking every day, and his explanation to Lincoln of why he does might just be rationalization. And though Charlie openly condemns the fast-paced lifestyle he once lived in Paris, we can't help but notice the tinge of nostalgia and longing in Charlie's description of the city and the good old days:
Outside, the fire-red, gas-blue, ghost-green signs shone smokily through the tranquil rain. It was late afternoon and the streets were in movement; the bistros gleamed. […] The Place de la Concorde moved by in pink majesty; they crossed the logical Seine, and Charlie felt the sudden provincial quality of the Left Bank.
Charlie directed his taxi to the Avenue de l'Opera, which was out of his way. But he wanted to see the blue hour spread over the magnificent façade, and imagine that the cab horns, playing endlessly the first few bars of La Plus que Lente, were the trumpets of the Second Empire. (1.26-27)
"It was nice while it lasted," Charlie said. "We were a sort of royalty, almost infallible, with a sort of magic around us." (1.44)
Here we see the part of Charlie that wants to return to his old life. This is one explanation for his leaving the address for Duncan at the start of the story. It also explains why he keeps tempting himself by visiting all his old haunts. If he were truly reformed, we suspect, he would avoid the scenes of his former crimes. You don't really expect to see a recovering alcoholic hanging out at his old bar, do you? So sure, most of Charlie wants to reclaim Honoria and start a new life and be a new man. But part of him wants to return to his old lifestyle. It is this tension that makes "Babylon Revisited" so darn good.
Ultimately, we can't say either way whether Charlie is truly a new man. The ambiguity of his transformation is central to the story; we're supposed to be unsure. If we knew how to read this text for certain, then we would know for certain who was supposed to get Honoria, what was supposed to happen at the end, what was just and what was unfair and who was right and who was to blame. And then it's not a terribly interesting story. Remember also that "Babylon Revisited" is rooted in Fitzgerald's own personal experiences and in the financial crisis of his time; blame is not easily assigned in either one of these spheres.