Hi Bette, I have a Bonnie Book called "Tut Tut Tales" (similar to LGB's), by Marjorie Barrows, illustrated by Clarence Biers. My copy says c. 1975 but was probably first published in the 1950's judging by the styles of hair and clothing in the pictures. It's actually a collection of six stories about different children with a little lesson each one learns. The crying little girl was named Imogene Mabie, who was a "cry baby"; the story went about like you remember. Hope this helps!
Does anyone remember a story about a little girl who cried all the time? She cried about anything and everything. One night she went to sleep and forgot to turn off her tears, and when she woke up her bed was floating in all the tears. She decided not to cry about everything anymore.