The song that 2 of the Hobbits sing in ‘Shortcut to Mushrooms’, is sung by Merry and Pippin in the movie (FotR, Extended Edition) There it goes:

Hey ho! To the bottle I go
To heal my heart and drown my woe
but there’ll still be many miles to go
Sweet is the sound of the pouring rain
And the stream that falls from hill to plain
Better than rain or a rippling brook
-Pippin-IS A MUG OF BEER INSIDE THIS TOOK!