As per the books, the events went differently.
Will was on a tree when the Others killed his commander Ser Waymar Royce. After Royce was dead and the Others left, Will climbed down. But once he was down, he was assaulted by the walking corpse (Wight) of Ser Waymar Royce. Will never survived the attack. It was Gared, the third guy they left with the horses, who escaped. Gared already had a feeling that something very wrong was afoot but he was ignored by his young Commander. Since he was at a safe distance already when the Others did for Ser Waymar, he escaped, aided by the fact that he had horses. There's no information on how exactly did he escape except that he was arrested south of the Wall and executed by Lord Eddard Stark.
As for Sam, Sam never encountered the Others like he did in the show. He was present at the Battle of the Fist of the First men where the Night's Watch escaped with the Lord Commander personally leading the cavalry charge which broke the undead lines. After that it was a long retreat back to the Wall. They lost many men to sporadic attacks from the Others, sickness and cold but they eventually made it to Craster's Keep. Once there, the mutiny happened and honest men made their own way to the Wall, Sam included. He did face one Wight (formerly known as Night's Watch man Small Paul) and killed him with the dragonglass dagger that Jon had given him but that was before the mutiny. Sam was attacked once more by Wights after the mutiny when he was escaping the mutineers with Gilly. Sam killed one of them but he was beset by more Wights. He was saved and escorted to the Wall by Coldhands.
So no, in answer to your question, in the books at least the Walkers don't deliberately leave survivors. Some people like Gared or the NW men at Fist of the First men, however, manage to escape with sheer luck or desperate courage.