If I recall, the Other killing is pre-Commander killing, because after the Commander is a fucking mess and Sam gets the fuck out of dodge. The dead Nights Watch attack them later at an abandoned village, then they eventually run into Coldhands.
Looks like you are correct.
When the Others attack the Watch camp at the Fist of the First Men, Sam manages to send word of the attack to Castle Black and is able to escape with a group of about fifty survivors led by Jeor Mormont. The march is hard on Sam and he is helped by his friends Grenn and Small Paul. Eventually they are separated from the majority of the men. Paul, who carried Sam for much of the journey, is attacked and killed by an Other, but the weight of his body disarms it as he dies. Samwell, who is carrying a dragonglass dagger that was among the cache found by Ghost, uses the opportunity to attack, and blindly stabs the Other, who dies instantaneously from the dragonglass blade. Killing the Other earns him the nickname "Sam the Slayer".[6]
He and Grenn rendezvous with the remaining survivors at Craster's Keep, where Gilly gives birth to a son. Shortly thereafter, violence erupts in the keep, with some Night's Watch men believing that Craster was holding out on the food he provided for the Watch, and others harboring mutinous intentions toward Commander Mormont. Craster and Mormont are both killed in the fighting, Before Mormont dies he gives Sam his dying wish, that Jorah, his son, join the Watch.[7]
Sam is forced to flee, joined only by Gilly and her newborn. They make it as far as a village Sam believes to be Whitetree but are then attacked by wights, including a reanimated Small Paul, who attackes Sam first. Sam fights him off and stabs him with the dragonglass dagger, but it proves ineffective. Desperately, Sam strikes him with a piece of charred wood, which catches him on fire, killing him. They run, but are beset by additional wights, and are saved at the last moment by Coldhands.[8]