They finished at the last possible moment. The publisher had cunningly closed his office that day and ordered all his staff not to receive the manuscript, so Dostoevsky went to the local police station, handed it in, and got a receipt. Thus his career was rescued. And then he and Anna got married.
It's fitting that this drama produced The Gambler (a great short book btw!), since the entire episode was a high-stakes gamble, and Dostoevsky a notorious gambling addict. The marriage was a long and happy one and his life stabilized after that. His books maybe got a tad less exciting.
Joseph Frank's biography is the best English source but he spends too many pages on this for a Google Books link to be useful. This one works though: https://books.google.com/books?id=1C1K-BnFGFIC&pg=PA51&dq=do...
Re serials: it's mind blowing that so many great masterpieces of the novel were written and published that way, and yet are so coherent, Dickens included. How on earth did they do it? The unconscious is a consummate planner.