Books that are written to music are just meant to be sung. Don't shy away from belting out these books. Even if you have a terrible or out-of-tune voice, your child(ren) love you and your voice. Rarely will a child notice how the voice sounds. Other than the cozy bond that grows from sharing a book (which is enough personally), singing the book helps to break down the words. When it comes time for you child(ren) to learn to read, it will be easier to decode the words if the words have been heard before. Singing helps to emphasize the syllables.
A few awesome song books are:
Over in the Jungle: A Rainforest Rhyme by Marianne Berkes &Jeanette Canyon (EP Berkes)
This Jazz Man by Karen Ehrhardt & R.G. Roth (EP Ehrhardt)
Nellie Belle by Mem Fox & Mike Austin (EP Fox)
The Babies on the Bus by Karen Katz (EP Katz)
Five Little Monkeys Jumping on the Bed by Eileen Christelow (EP Christelow Counting)
Rock-a-Bye Romp by Linda Ashman and Simona Mulazzani (EP Ashman)
Down by the Barn by Will Hillenbrand
Hickory Dickory Dog by Alison Murray (EP Murray)
Jane Cabrera has a slew of traditional songs with new verses. Her books are found in the EP section under Cabrera. Jerry Pinkney has done Twinkle, Twinkle, Little Star with all the verses (! A must!!) and also The Three Little Kittens. In the board books, check out Annie Kubler books and Marla Frazee's Hush Little Baby.
With these traditional song books, feel free to sing as you know. Don't be constrained to the printed words. For instance, I sing Hush Little Baby with a different wording from the ones Marla Frazee chose. I just sing it the way I know. (For my story time friends, you know I "misspeak" quite a bit during our songs or rhymes; it's okay! What you care about is not perfection, but the joy from doing the songs.)