I started Uni with extreme social anxiety and depression. Granted, I didn't have so much on my plate; my parents are both relatively healthy (I really do hope that everything goes as well as possible for you!), but my family life still was (and isn't) great. My parents have been threatening divorce pretty much since I was born, argue at the top of their lungs every waking hour. My mother was the extreme coddling type, which lead to my social anxiety and lack of adulting skill; as you say, I still often don't feel as though I can function as a "proper" adult, I seem to be quite naive compared to others' my age and I enjoyed my comfortable home life far more than was healthy. My dad was distant and we spoke maybe once every other day or so.
I was very sure that I wasn't going to do well socially at Uni. The thought of having to socialise weighed on my mind a good 5-6 months before I'd even started. In secondary school, I'd struggled even to ask the person sitting next to me for a ruler; I couldn't imagine making friends that easily. As generic as this piece of advice is, it's the truest thing you'll hear about Uni: everybody is in the same boat. I think that when you get to Uni, you'll find that the atmosphere is very different. EVERYONE wants to talk and get along. Where throughout my school years, I was painfully reserved, never spoke unless spoken too, started sweating when asked a question etc, during the first week of Uni, it was as though all of that had alleviated. I could talk to people very easily and now I find myself in a stable group of friends.
I think it's just a matter of time, and going with the flow. Once you get to Uni, you will open up quite naturally. But you have to make sure that you take the opportunity. Don't sit there FEELING like you could talk to the person next to you and not actually do it, else you're pushing yourself further into a hole. Just start with a quick "Hi, what course are you doing? Are you a first/second year?" etc, and trust me, it WILL go well.
As weird as it sounds, I think that you are in a good position right now, moreso than most other people. I'm sorry for what you are going through, but you'll need to start anew many, many times in life and you're in a good spot right now to do so. Many of my friends had so much drama and baggage to carry on with them through Uni. Right now, it's a blank slate. You can do whatever you want with your life right now. Uni is the perfect place to do that. I drifted from my friends too, to this day I have little to no contact with my secondary school buddies, but that in itself opened up so many doors for me at Uni.
The thing about Uni is that really, we're all kids here. Hardly anybody has a full time job, many students are still depending on their parents for funding. What's different about how we live compared to how we did in secondary school? We've moved out and we now have to cook for ourselves, but in the end we're still studying for another 3 years and although I am personally not into that scene, going out and partying. I think you'll find that your existence and who you are as a person doesn't change when you go to Uni. You can still be carefree and enjoy the things that you used to. It's just that your outlook broadens.
As for your depressive mood, I have been in and out of therapy for a few years and nothing has helped me either. Believe it or not, a single good workout can really help. The first time I hit the gym at Uni and did a proper, sweat-inducing work out, I walked out of there feeling confident. It was a very chemical thing. Weird, but it works magic. When you go to Uni, you have the opportunity to do so many things of your own accord. Motivate yourself, get a gym membership, learn how to skate, whatever you want!
Lastly, I've lived by the fact that letting go is more often than not, an amazing thing!