I have social anxiety and I'm currently in university, so I totally get where you're coming from... especially the feeling awkward being alone in places. I have no friends at uni and people generally avoid me. Even staff members seem to generally dislike me.

I'll post the things that have helped me in case it might help you... Changing how I think about my environment has greatly helped me punch the anxiety monster in the face.

When you're walking around campus, look at the people around you. There are bound to be people who are alone, just like you. Do you think any less of them?

No one is looking at you sitting alone in the cafeteria and likely thinking anything other than, "do I know that person?", or "man I'm hungry that person's food looks great". Honestly? Everyone is really busy and involved in their own affairs, so no one cares. They aren't walking around looking for people they think are losers... in fact them assuming that you're a loser because you happen to be alone is not only false, but a very bad assumption. You could be enjoying a quiet moment of solitude. You could very well have friends that don't have lunch at the same time or are off campus. In other words, anyone who is thinking negatively of you eating alone or going to the gym alone or doing anything alone is not very perceptive and is being completely unreasonable.

Every time I feel awkward in public I remind myself that it's very likely no one gives a flying monkey what I am doing or what I look like.
If someone does happen to care and is judgmental, they are probably not going to say anything anyway. Those people are terrible judgmental people that you wouldn't want to know though, so no loss!

When I feel anxious, I just turn to that reasoning and I feel much more confident. It took a little while but as I got older I have become better at it.

Joining some kind of activity that you enjoy will help you meet people on your own terms (you may be different, but personally I don't like random people approaching me). Activities give people great excuses to talk to other people because you're starting with a common interest. It's low threat.
This has been something that has helped me off campus more than on, because my interests are really limited and most not offered on campus... but that's another story, not important here.

...and living on campus is not for everyone. I never tried because I knew I would hate it, and an acquaintance of mine that tried had to go home because he hated it so much. It's not due to any personal failure, I think a lot of people are just not suited to live in such a stressful, unpredictable environment. In fact I think the people who can handle the social pressure and chaos (and unclean bathrooms) on a daily basis like that while juggling lack of sleep, study, work, etc... are the abnormal ones.