What happened to Sam?

You actually just refutted yourself with that quote. It says they had the choice, "die at their own desire and of free will".

Orofacion: I may be wrong about this, but I always took that to mean that, once they reached Aman (or Tol Eressëa, if you prefer), they were given the privilege of the Numenoreans and their descendants, to choose the time of their death. If they refused to make a choice, as their span of time drew to a close, they would, as Aragorn said in App. A, "..wither and fall..".

Anyway, that's my perspective.

Máme