Here's an improved version. It occurred to me that the big jumps in the B part are for yodeling. Yodel softly - you don't want to wake the baby.

[A ]Evening shadows are falling; the west is still red.
Little eyes are so sleepy; come rest your head.
Bold Buster keeps monsters from under the bed
as I'm rocking my baby to sleep.
[B ]Close your eyes now and listen. Hear the whippoorwill.
He warbles by night and he hides by day.
Count him: one-two-three, one-two-three, silly whippoorwill!
You can't chase that old moon away.

[A ]Little froggies are peeping at Mockingbird Creek,
Saying "Night-time is coming, it's time for sleep.
Daddy enters so gently, in stockings he creeps;
for I'm rocking my baby to sleep.
[B Counting one two three four five six; hush, you whippoorwill;
serenade us from some far and moonlit hill.
Little baby is weary; now don't you make her/him weep.
For I'm rocking my baby to sleep

[B]oh deh lay hee, deh lay hee, oh deh lay hee oh
Pretty birdies go hide in the woods so deep.
].oh deh lay hee, de lay hee, oh deh lay hee oh
while I’m rocking my baby to sleep.
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If you think that froggies and birdies are twee, take a few deep breaths. Next imagine that a baby's head is on your shoulder, surprising heavy and beautifully warm. Also small and vulnerable. Once you do, singing of froggies and birdies will seem completely natural.