About feeling socially awkward
You mentioned that you're uncomfortable telling the other person that you have social anxiety. That's perfectly understandable. Although the other person might very well understand, and even respect you for your candor, many people find it uncomfortable to discuss their anxiety with others, and there's no reason you should do something that makes you even more uncomfortable.
I would, however, suggest that you not think of your anxiety as a "social failing," because it's merely a characteristic of your personality, not a failing. You might find enjoyment in the book Quiet: The Power of Introverts in a World That Can't Stop Talking. It points out the value that people who aren't naturally outgoing bring to their environment.
I would suggest keeping in mind that when another person draws attention to shyness, they don't necessarily mean it in a demeaning way. If they are outgoing themselves, they might not understand why someone would be shy, and their willingness to draw attention to it is a natural outgrowth of their own inhibitions and willingness to speak freely. Personally, I have been called shy by people who also expressed respect for me in other ways, so even though it took me off guard to be called out, I was able to take it without feeling like it was an attack on my personality, or an undesirable characteristic. Of course, that doesn't mean handling these situations is simple, for me or for anyone.
What you can do when someone draws attention
Most people do understand that everyone is different, so if you're seeking a middle-ground between acknowledging the anxiety and trying to fake it, or pretend that the anxiety doesn't exist at all, there is the option of acknowledging what they say in a minimal and honest way. You can always say something along these lines:
That's just the way I am sometimes.
I'm just being myself.
That's just me.
If you try something along these lines, you might find that the other person brushes off the issue. They've met shy people before, and they probably won't push the issue further. If they do, and continue to needle you for being shy, then they might not be the sort of person you want to be associating with anyway. You could tell them, "Sorry, it's nothing personal," and allow them to either let it go or else you don't have to keep answering their questions. You're not obligated to be interrogated by other people because you feel uncomfortable.
Easier said than done
Out of the pieces of advice people receive for handling awkward situations, few of them are easy for a socially anxious person. People often say, "laugh it off," but sometimes laughing or grinning can be difficult during moments of anxiety, and attempting to fake it when you aren't really feeling it can cause more discomfort.
Similarly, brushing off a comment by saying, "That's just the way I am" might not seem easy at first when in the heat of the moment. Maybe try this sort of response the next time you feel as though you're in this situation, and consider it practice. If it doesn't feel right, you can either decide to keep practicing, or try a different approach the next time.
Either way, overcoming specific aspects of social anxiety -- as with other anxieties -- is often accomplished through immersion, or subjecting oneself to difficult situations until they become easier to handle. Think of each time that this happens as another step toward feeling more natural the next time.