How do you place a judgment against someone if they moved out of the state AFTER you won your case and you no longer know where they live?

You could try calling family members or friends if you know of any, and give them some believable reason for contacting the person, like you owe him/her money or you borrowed something and want to return it. There is the option of hiring a 'skip tracer' but that would result in more expenditure that you probably wouldn't be able to recover. Unless you can get the help of a neighbor or family member of theirs it will be pretty difficult.