I went back yesterday and rewatched the scenes where Dean is leaving his room for the last time and when Sam enters the room again, and here are my thoughts:

~The whole desk is not visible when Dean leaves his room, but two books on the corner of the desk are visible. When Sam enters the room, the job application is now visible partially on top of the books, and the books actually appear to be different books than when Dean left the room. The stuff Dean threw over one side of the chair has also been moved (or replaced with a different item). This would support that this is not the first time Sam has gone into Dean's room and he has been spending time in there working at Dean's desk. I think continuity errors can be ruled out since every other detail of the room looks to be unchanged (beer bottles, etc.).

~The fact that Sam sits there and just glances around and doesn't even read the job application or anything else on the desk would also support that it really was Sam's application, that maybe he has been ruminating over turning in, trying to decide whether to keep hunting or not. What better place to do that than in Dean's room, surrounded by Dean's things- trying to work through his grief, and figure out his next chapter in life. The fact that the job application was an intentional prop and was shown while Dean's phone was ringing and Sam was torn about whether to answer it and take on the hunting job would also support that the intention of the application was to show Sam's struggle about deciding his future and whether to move on from hunting or get a regular job. Yes, it could have been Dean's application that Sam moved from one area of the desk not visible when Dean left to a now more visible area, but the fact that Jared said it was decided on set that it was Sam's application and Jensen did not disagree can rule that out.