Welcome to the Downs,
stryfe!
I did initially feel the same way you did, but then I found that The Lord of the Rings never really ends. No, I don't mean metaphorically. If you dive into The Silmarillion, The Hobbit and other works by Tolkien, you'll see that there's such a huge world that there's no way you can think it to end after Frodo leaves. At least, that was my experience.
I suggest you read more of Tolkien's works, and you'll probably find yourself compensated for the bittersweet ending in LotR
