You're definitely not alone in this. I felt exactly the same way you did. Then again, I think it just made me love the story even more. There was enough of fairy-tale happiness towards the end of the book – an ending that was cliched and sappy would, in my opinion, have killed the whole majesty of the story.
Yes, I was very sad when Frodo left, but after discovering The Silmarillion and other works regarding ME, I felt pretty elated. You can just bury yourself into any story you want.
And of course, the discussions in the Downs never lets it end