Sandy and Caroline Paton have a record (or used to) on Folkways called "I've Got a Song." It's terrific.
Sharon, Lois & Bram are a Canadian trio whose first two albums were especially good. I think the first one might have been called "One Elephant, Deux Eléphants."
"Free to be you and me" was a television special put together by Marlo Thomas in the 70s. Don't know if the record or songbook is still available, but they had charming songs and stories read and sung by people like Alan Alda, Rosie Grier, Carol Channing, Dan Smothers, and more.
Barry Louis Polisar is a children's entertainer who writes songs with titles like "Never Cook Your Sister In a Frying Pan," "You Can't Say Sssppplllbbbhhhh on the Radio," "Sally Eats Shoelaces, Straw and String," and "They Said Eat the Broccoli."
I don't know how many, if any, of these are still available, but they're pretty terrific. You might also try your local library for historical records such as Leadbelly singing with children, Ella Jenkins, Malvina Reynolds, and so on.