Hi there. I'm just going into my second year at university, and found my first year pretty difficult too. So you're not alone. Lots of people feel this way. I know you must feel extremely isolated, but when you picture that many people feel lonely together, it can be somewhat comforting.
My first piece of advice would be to have patience with yourself. Remember the saying 'slow and steady wins the race'? Don't feel obligated or pressured into socialising. Check in with yourself regularly each day to see how you feel. If you'd prefer to have a quiet night alone in your room, there's absolutely no problem with that. Self-care is important here, and so you need to make sure you're eating properly.
Not eating will only make things worse. I also skipped many meals as I didn't want to be in the dining hall alone (at first). I soon learned to not care what anyone else thinks. I know it's tough, but try to focus on one thing and one thing only; the reason why you're in the dining hall. Yes, people go to socialise, but your main objective is to eat. So make sure you do. Go on your phone if you have to, so you don't appear as noticeably alone.
I hated the icebreaker events. I was already an outsider as I was a state-schooled northerner surrounded by posh southerners who had been to private school. I don't go clubbing and the dining hall was also a bit of a minefield too, at first. I found my first friends in the dining hall queue. I was terrified, but managed to approach them, introduce myself, and ask to sit with them. They agreed without hesitation. This may seem daunting, but trust me, taking the 'risk' is worth it. With anxiety, you're essentially putting yourself in a cage and locking it, but you're the one holding the key. That's not to say that the way you are feeling is your fault at all - there will be an explanation, whether you know it or not, for why you feel the way you do; a series of experiences, maybe. Signals are being sent to your brain, the amygdala, and you are going into fight, flight, or freeze mode. It sounds like you're almost always in flight mode, and veer towards avoidance. I can relate to this a lot, so again, you're not on your own.
Secondly, don't kick yourself. Your entire post is filled with self-loathing and unnecessary criticism. Stop getting into the mindset that you are forever damaged, weak, or stuck like this forever. You are none of those things. You even said yourself that at one time you started to actually make a bit of progress. This is really positive and something you need to remind yourself of. You are capable. You are worthy.
But just think about it. Even if you had managed to take a few tiny steps out of that cage, no matter how mentally exhausting it was, if something miniscule went wrong, I guarantee you'd just be kicking yourself back into that cage again. It's completely habitual. You become anxious, and then depressed. I constantly found myself in this state. So my point here is to be gentle. I know it's difficult, but even just taking the time to remind yourself of your worth and your good qualities every morning could help.
Here's a thought - if you saw someone standing on their own, surrounded by a few groups of people in conversations, would you think any less of them? Would you think they deserve to feel humiliated?
If your answer was no, then, again, why are you beating yourself up so much? Is it worth it?
If you are feeling anxious, acknowledge any negative thoughts that are contributing to this emotion, and then try your best to let them go. Fighting with them isn't going to help you. It'll just drain you. Mindfulness could help, if you haven't already tried it. I like the app HeadSpace, but there are others.
Baring everything else in mind, it does sound as though you want to fight your anxiety and become less anxious in general. Obviously, as I mentioned before, take your time. But when you feel ready, try to put yourself out there. You don't want to be locking yourself away physically as well as mentally. This will only manifest your thoughts and make them seem even more real.
I would recommend speaking to your psychologist about the social side of things and how much you struggle. Social interaction is unavoidable at university, as you'll know by now, and so you may not have thought much of it when you were at home. You've had a shock. The good part is, you can start to recover from this and get into a routine.
You may feel that you're trapped in a vicious cycle right now, because unfortunately, this is just the product of anxiety. You feel anxious, you struggle socially for it, and then you beat yourself up for it. You can overcome this and learn to manage your life.
While you cannot help feeling socially anxious (I would advise you seek support for this), you can start to work on how you react to this and how you treat yourself as a result. Putting yourself down is only going to make things worse. Try doing a bit of reflection. What advice would you give to a stranger if they were in the same position?
May I ask if you're on any medication, such as antidepressants or beta-blockers? Have you ever tried any of these medications or similar?
I'm just going off my own experiences with mental health and cognitive behavioural therapy, as well as psychological research.
I hope this is of some help.