I mostly lurk here because I'm not in CS (I am trying to learn programming after having enjoyed a successful previous career) but I recognise social anxiety so I'll pipe up.
So, I am coming off about 15 years working in offices in the business world. In everything from small (<50) medium (>100, <1000) and large (>10,000) firms.
I identify with many of the behaviours you describe. It was interesting to also note how much internal self-discounting you are going through, to remember I too felt that at the outset of my professional career. I had completely forgotten that, but felt it keenly at the time.
A couple of things. First of all, while I am not a psych, it wouldn't surprise me to learn that social anxiety exists along a spectrum. While I would encourage you to seek and get treatment, I would also suggest that you recognise that your functional abilities actually sound pretty good. From the sounds of it, you find social interaction taxing, but you can do it when required. That doesn't sound to me like a debilitating case such that you actually can't leave the house. I don't think you should be discounting yourself as being out-of-the-running because you're way off the charts. From what I've seen, and my own personal experience of the condition, you're not.
That's one reason not to discount yourself, which is looking at it from the perspective of your condition.
The second reason not to discount yourself is from the perspective of the work environment you'll be entering. I found the social pressures of work to be much less than at university. At university, there was a strong focus on shared social activity or making friends, or whatever ... a lot of it part of peoples' identity formation. At work, I found all of that had just about fallen away. Remember, you're entering a situation of people who are at (and from) immensely diverse stations in life* (*mandatory Silicon Valley start-up exemption here) - your bosses might be in their 30s, 40s, or 50s, be married, have families ... It's all vastly different from the social pressures of a single cohort of 20 year olds all establishing their relationships to one another at college.
Apart from that diversity of people, there is also the fact that you're there to, well, work. Again, this was a big change for me and was a pleasant surprise, strange as that sounds. I absolutely thrived in my professional career unlike at university, because I could talk to lots of different people about work. There was magnitudes less shit talking in a professional environment than there was at uni. Sure, you might top- and tail- the occassional conversation with personal stuff - but if you can get through the "How was your weekend" convesation. it's pretty much straight down to business. If you can get peoples' confidence that (a) you know what you're doing, and (b) you're reliable, such that (c) you deliver the goods, and (d) you're not an actual asshole to people such that they positively want to avoid you, then you are off to the races.
Seriously - you may surprise yourself here, in that work will become a place where you are getting repeated validation from people that you are good, you are valuable. You'll not fail to notice this --- People will select you for projects, they will want you on their team, etc. This is really good validation that you just never really get in college. And none of it is because how good you talk shit or how well you dress or what other conventions and filters of success were applied in the very narrow bandwidth of 20 year old sophomores.
Paradoxically, that then actually does put you in a situation where you might enjoy going for a beer of a Thursday or Friday night with your colleagues. Especially if you've gone through a project or some other deliverable. There's always a good feeling de-briefing over a cold beer or three. And you can still go home before the DJ comes on if you like. But it actually will get you out and about on more occassions, and in more comfortable occassions, than you might find if you're not in this environment, because you'll at least be "talking shop", something that should be more comfortable than talking shit.
I'm not trying to paint office life as all roses - you are too intelligent to believe me if I did, and anyway, it's just not. There's politics and all that stuff, and that gets a bit messy. But addressing the fairly narrow field of concerns about your condition, I can positively say from my experience, with similar anxieties at the outset of my career, that work on balance was a very positive experience for me.
So, to wrap up, I would encourage you first of all, while you should seek out professional advice, please don't turn your condition into some big thing that it's not, and second, don't discount yourself based on preconceptions of what the workplace is like and how you're going to fit in. It can be hard work but if you're fundamentally good at what you do, and a capable person, it could be very positive.