I feel the same way. I often feel an ackwardness when around too many people, I seem to loose all train of thought and find it very difficult to make conversations, I am too preoccupied on different things, such as what they think of me, why a they looking at me the way they are and how am I going to get out of this situation.

As for claustrophobia I do believe that there is a link between claustrophobia and social anxiety. I think many people believe that claustrophobia is purely a fear of small spaces, such as lifts, caves and small rooms, but they forget about the situation like talking to people and feeling as though you are suffocating in a situation that doesn't have to be physically confined but internally confining.
I was looking online and found this info about how there is in fact a link between the two.

"Causes of Claustrophobia
Claustrophobia is actually generally considered a symptom of anxiety, rather than its own anxiety disorder. This is likely because most people that develop claustrophobia already have at least some type of mild anxiety. Take my anxiety test to find out more about your anxiety.

This makes some sense too because several disorders seem to be affected by or play a role in claustrophobia:

Generalized Anxiety Disorder – It seems that generalized anxiety disorder seems to play a role in the development of claustrophobia. Most likely there is some correlation between what led to a person developing the condition and what leads to the development of anxiety
Panic Disorder – When someone is in an enclosed space, they become more likely to suffer from a severe panic attack. Chances are that the individual was at greater risk of panic attacks or panic disorder, which means that they already showed at the very least a genetic pre-disposition tot the condition.
Social Phobia – Those with social phobia often experience a feeling of needing to escape as part of the condition, although generally it's not limited to enclosed spaces. It's possible that some of this fear ends up tangentially passing over to the way they feel in any small space."

Hopefully this is helpful in someway