How do I manage this interview while dealing with this problem?
The key to getting better at anything, or getting past this sort of problem, is practice. You get better at sports by practicing. You get better at typing by practicing. Interviewing is no different - you will get better (and more comfortable) with practice.
Start with a trusted friend or family member. Have her/him take the role of interviewer and ask you interview-type questions. You answer, while remaining calm, maintaining good eye contact, etc. When each session is done, ask your friend how you did, what you didn't do well, and what you could do better. Repeat several times until it starts to feel more natural.
You could then have your friend find someone you don't know to take on the role of interviewer and practice some more. If you are still in contact with your university, often their placement offices can set you up with someone who can help you practice your interview skills and offer interview coaching as well.
Some folks even video-record their practice interview sessions. Playing them back can often be insightful.
To be clear, the problem maintaining eye-contact was just an example.
The point was to make it clear that I simply cannot function around
If there are medical issues preventing you from actually functioning around other people, you need to address those first.
It won't make any sense to find some trick that will get you through interviews, only to be unable to function around other people once you are on the job.
If it's just a matter of nervousness or inexperience, you may just need to suck it up and force your way through it until you naturally get better at it over time.
If instead this is a medical or mental health issue - seek professional help first.