Hi,
Regarding your lullaby, My Granddad used to sing this to me at bedtime during the Late 50,s early 60s.and I sing it to all my grandchildren
I think you have written the main of the song, but my granddad
used to sing before it;

Gentle moon that gives us light,
Through the long and stormy night,
Will you kindly tell us pray
Where you go to in the day.

In your Starry nest,
Do you ever rest
?
Goodnight mr moon, we shall..........
and you know the rest, hope this helps