The lyrics I remember my grandma singing are:
{Not necessarily in this order}

Was in the merry month of May
And the willows they were blooming
Sweet William on his deathbed lay
In love with Barbara Allen

Sent a messenger out of town
To the place where she was dwelling
My masters sick and very sick
If you please Ms Barbara Allen

Slowly, slowly she got up
And slowly she went to him
And all she said when she got there
Was young man I think you're dying

Oh yes I'm sick and I'm very sick
And death is very near
I'll never see my time again
If I don't get Barbara Allen

Oh mother dear, go make my bed
And make it long and narrow
Sweet William died for me today
And I'll die for him tomorrow.

This was what we sing to our babies in this family. The tune is soothing I guess. If anyone knows these lyrics let me know.