We had this record too. It was in a compilation of children's songs... my sister, who keeps everything, may still have it:

A strange little bird is a weez wump
it ends where it ought to begin
it eats only moss from a moonbeam
and sleeps on the point of a pin

it's wings are too curly to fly with
so it floats on a leaf in the breeze
it takes daily baths in a dew drop
and nods to it's own melodies

A weez wump is nice, but it's lazy
It won't build a nest of it's own
It hides from the rain 'neath a daisy
and cries when you leave it alone