My six year old WAS the same way. Like the other responding mother's child, mine is shy, doesn't want to be wrong, and really likes to please. Now we cannot get him to talk quietly; he even yells, a lot! I think he competes with or joins in with his siblings for attention. So, first, continue in your efforts with patience and long-suffering.
Also, my six year old is receiving speech therapy. Your affluent, enunciating, three year old may not qualify for services, but you can ask a speech therapist for advice and ideas on this issue. So, secondly, ask your pediatrician with whom they would recommend you to speak. Depending on your state's services for Early Childhood Development,there should be an agency. Perhaps, you could ask well-trained preschool staff who are sure to have such contacts. Our state university children 's hospital works with therapists as part of their follow up care for neonatal patients who themselves are familiar with local agencies. So, secondly, if you have access, check out what the professionals say. Even a preschool teacher at least, has experience.
Being mothers of three, we know that each child is unique. Your three year old sounds just like my second child who is also three. When I remind him to speak up he does. So, I let it go at that. I'm gathering yours doesn't always respond to what you are trying. What is going on for her emotionally? Is she aware of the new baby? How does she feel about being a big sister again? When you remind her to speak up are you annoyed or interested in your tone of voice? Also, would I be very tired and exasperated from raising, training, and caring for two young children, while also teaching college students, and growing a new baby? YES! So, thirdly, my latest offspring IS teaching me to let go. It has been unpleasant at times, but I want to be truthful and real to you.
Just a sidetrack, I thought of something else you could pursue: I read about siblings' speech being affected by each other; is your 19 month old speaking? Is he making noise? Is he quiet or loud? Lastly, these things are complicated. So, give yourself and your daughter time; seek more help from professionals, I'm wishing and hoping someone will respond to you soon with practical and tested ideas for you to try; and I have been on the receiving and giving end of this issue, as a shy, soft spoken child grown up to be harried, busy mother of three, but aspiring to be of character: patience, loving-kindness, so, I think I can say: use this as an opportunity to develop your mother-daughter relationship which you obviously cherish.
My 13 month old daughter is stirring up in the family bed, and with it being flu season at my house...the sick need their rest, the healthy have their needs, too, and I'm stuck in the middle of them all! (Not literally) What was I saying about letting go? Does this mean I'm not Super Mom in that area to? But that's my request story, isn't it!?! Thanks for listening, God bless you, and good luck!