The answer largely depends of whether you accept the significant amount of evidence that Chuck Shurley was not a normal prophet, but in fact, that
Chuck Shurley was God.
The evidence here includes statements from Eric Kripke and Rob Benedict (the actor who played Chuck), so IMO it's pretty strong. It was never confirmed on-screen, though, so some people dismiss it.
If you believe that theory, of course, what happened to him is obvious: he had done what he needed so he left.
If you don't believe that theory, and assume he's just like every other prophet, then the premise in your question is a bit off. Prophets are not angels, so they are not possessed and don't have vessels. Prophets are born that way, and when one is needed they are called up from the group of waiting prophets. So, nothing happened to the Chuck Shurley from "before" he was a prophet -- there was no "before".
Unfortunately, Chuck Shurley's time as prophet worked very different from Kevin Tran's, so it's hard to say exactly what happened to him. (This, BTW, is a big part of the evidence for the above theory.) As far as we have been told after season 6, prophets maintain their role until they die, when the next one is called up. What this means for Chuck is unclear.
If the prophets in Supernatural are supposed to be in the same line of prophets as those from the Old Testament, like Isaiah and Jeremiah, most of those are generally assumed to have lived long lives and died of old age, but they continued to be prophets until the end.