First off, the one Ernestine I know of is Ernie's little niece. So Scarecroe, if you have some other information as to a second Ernestine, please share with the rest of the class.
Secondly, there are two sketches I know of where Ernestine appeared.
1 Ernie has Ernestine with him and tries to teach her to say his name. He keeps at it, even breaking down his name into two syllables and banging two small blocks to get his niece to say "Er-Nie." But no matter what, she just won't say his name. Then Bert enters the room and closes the door behind him. We hear Ernie's classic greeting of "Hey Bert!" and then we promptly hear Ernestine say excitedly "Bert!" Guess she inherited some of her uncle's ability to frustratingly annoy others.
2 The other sketch involving Ernestine was one where Ernie tries to find similarities between the two of them and describes what she looks like, sort of. He says that these are the following differences.
Ernie: I'm a boy and Ernestine's a girl. I have dark and funny hair, and she has light (blonde) and curly hair. She wears a bonnet and I don't wear a hat. I'm wearing a striped shirt and hers is plain (probably light blue or yellow). And of course I'm big and she's little. So no, I just can't find many things that make us the same, but I still love her. And then he laughs his Ernie-laugh, followed by Ernestine who lets out the same laugh.
3 There's probably a sketch with Baby Ernestine, Ernie's niece, and Lily Tomlin's character of Ernestine the Telephone Operator, but I'm not sure as to such a sketch existing nor what happened should it exist.
Third, there's a picture for people to color in of Ernestine in the coloring book, the New Who's Who on Sesame Street which I have as part of my collection.
Hope this helps and have a good day.