Having done rescue for almost 18 years - the first thing we ALWAYS do is change their name. As you said, if you use their previous name they will associate it with their previous life. It is simple - new name = new life.
A lot of people think that you cannot change the name if they are older than a couple of months old. But we always change names. Sometimes because we have no choice. It isn't like you meet them at Animal Control and they say "Hi, my name is Buddy". But in cases like yours, because their old name was associated with a not-so-nice past.
My white boy came to us when he was 11 months old and his name was King. We quickly renamed Loki and have never had a recognition problem. Actually, my husband was talking to him a couple of days ago and mentioned the name 'King". Loki cocked his head, looked at my husband, gave a small bark, and then a low growl. Too funny. Obviously, he still hates that name and situation.
Enjoy your new girl. She will be the love of your life - right up their with your husband. And I still haven't figured it out but she will always remember that you 'saved' her and chose her. And she will thank you for it everyday for the rest of her life.
As a rescue person, I thank you for taking in a puppy that needed a new life and family. Being an older puppy can make it hard for her to find a new family. You will not be sorry that you did - I promise you that.