Can a 17-year-old move out without being legally emancipated?
With Parental consent, I believe so. However "emacipated" is not the term you are looking for, it only gives you the ability to sign into contracts, car payments, credit card, etc.
It means being set free. In the case of a teenager being emancipated from their parents, it means they are legally considered an adult and can make their own decisions and live where they want to. In the case of slavery in the United States, the Emancipation Proclamation "emancipated" the slaves; it set them free from slavery.
Can your 16 year old move in with her 27 year old sister and get welfare on her case without being emancipated?
No. Being pregnant/giving birth does not give a minor any additional rights. They are still subject to the control and authority of their parents until they reach the age of majority (18) or they are legally emancipated. And by the way, being pregnant makes it much more difficult to convince a court to emancipate you.