So why does Sadie not like Sophie?
Well, you don't know what Sadie's past background was like, in terms of her experiences with other dogs. She may become easily defensive and aggressive because of those experiences, or she may not have been well socialized to be around other dogs. And Sophie's initial enthusiasm and "welcome" might have frightened her. In addition, I don't know what kind of surgery that Sadie just had, but she might be in discomfort and feeling rather on edge because of that, coupled with being in new, unfamiliar surroundings.
Don't panic. I don't think what you're describing is at all that unusual. I really think the problem will work out with time, familiarity, and some obedience and aggression training for Sadie.
Fortunately, your dog Sophie seems to be more accepting of the new dog who has suddenly appeared on her turf--probably because she is used to having foster dogs in the home.
Continue keeping both dogs on a leash, and apart, when you bring them together in the same place--you shouldn't put either dog at risk, and you shouldn't risk getting bitten. Try giving Sadie little treats when she remains calm in Sophie's presence, even for very short periods, and tell her, "Good calm". Don't let her get close to Sophie at all, until she is completely calm (and so is Sophie) through several of these "meet and greet" sessions. You might consider using a harness on Sadie with the leash, rather than attaching the leash to her collar, it's a firmer, less painful (for her) way of holding onto her, and keeping her body away from Sophie, if she starts pulling on the leash.
I think you have to start obedience training with Sadie ASAP. She has to learn to respond to you, and your commands, now. She needs to learn "sit" and "stay" so you can control her in Sophie's presence. She has to learn to inhibit her aggressive behaviors, certainly toward Sophie, and she has to learn how to calm herself, with your help. She's capable of learning all of that.
There is plenty of advice on the internet for dealing with this sort of problem. Here's one site, but there are loads of others.
There are also You Tube videos that address the problem.
You might also consider taking Sadie to some puppy classes, even though she's not a puppy, or to some regular obedience classes, where she'll be around other dogs while she's learning.
Given time, patience, firmness, and consistency, in training Sadie, and in helping her to adapt to her new surroundings (which include Sophie), I think it will all work out all right. Hang in there.
Congratulations on the new addition to your family.