Did the Buddha have advice about dealing with social anxiety?

Anxiety from loneliness is overcome with concentration; with developing calmness.

Anxiety from worries about social conformity is overcome with clearly understanding of right & wrong, i.e. morality.

The Dhammapada says, for example:

316. Those who are ashamed of what they should not be ashamed of, and are not ashamed of what they should be ashamed of — upholding false views, they go to states of woe.

317. Those who see something to fear where there is nothing to fear, and see nothing to fear where there is something to fear — upholding false views, they go to states of woe.

318. Those who imagine evil where there is none, and do not see evil where it is — upholding false views, they go to states of woe.

319. Those who discern the wrong as wrong and the right as right — upholding right views, they go to realms of bliss.

From AN 7.68, we find what is meant by one with a sense of social gatherings:

“And how is a monk one with a sense of social gatherings? There is the case where a monk knows his social gathering: ‘This is a social gathering of noble warriors; this, a social gathering of brahmans; this, a social gathering of householders; this, a social gathering of contemplatives; here one should approach them in this way, stand in this way, act in this way, sit in this way, speak in this way, stay silent in this way.’ If he didn’t know his social gathering—’This is a social gathering of noble warriors; this, a social gathering of brahmans; this, a social gathering of householders; this, a social gathering of contemplatives; here one should approach them in this way, stand in this way, act in this way, sit in this way, speak in this way, stay silent in this way’—he wouldn’t be said to be one with a sense of social gatherings. So it’s because he does know his social gathering—’This is a social gathering of noble warriors; this, a social gathering of brahmans; this, a social gathering of householders; this, a social gathering of contemplatives; here one should approach them in this way, stand in this way, act in this way, sit in this way, speak in this way, stay silent in this way’—that he is said to be one with a sense of social gatherings.

In Iti 21, the Buddha explains the benefit of having a confident mind:

“Here, bhikkhus, some person has a confident mind. Having examined his mind with my mind, I know that if this person were to die at this time, as if carried there he would be placed in heaven. What is the reason for that? It is because his mind is confident. It is because of the mind’s confidence that some beings here, when the body perishes, are reborn after death in a good bourn, in a heavenly world.”

Here's an excerpt from the essay by Thanissaro Bhikkhu on "Social Anxiety":

It's interesting that Ajaan Lee focuses on what other people say as one of the tests for a mind that's really at peace. The Buddha makes a similar point in one of the Dhammapada verses. "If, when other people say harsh things to you and you don't reverberate — like a cracked gong — that's a sign that you've attained true peace of mind." This might seem strange. Why does the test lie in how you react to what other people say?

The mind is very sensitive to this issue. We learn very early in our lives that our happiness is going to depend on how other people treat us. As children, we're surrounded by people a lot more powerful than we are, so there's always a sense of fear built into our relationships to the people around us. We become sensitive to other people's moods, sensitive to what they might do, what they might say. As a result, our center of gravity is placed outside because we're afraid of them, and we try to put up a wall outside ourselves to protect ourselves from them.

What this means is that our psychic center of gravity gets moved outside the body. If you've ever taken any martial arts classes, you know that if your center of gravity is outside your body you're in bad shape. You're in a weak position.

Now the Buddha doesn't say to ignore other people and just be very selfish. He says there's a different way to approach the whole issue of happiness. In other words, you find a source for happiness that doesn't take anything away from anyone else, so you don't have to be afraid of other people. When you're not afraid of them, you find that you can actually be more compassionate to them. So developing and maintaining this center inside is not a selfish thing. The Buddha's not teaching you to be insensitive. He's just saying to put yourself in a stronger position and to trust that you're stronger by not trying to go outside and fix up people's moods and all the other things that we think we can do with other people when we're dealing with them. Just stay inside and have a sense of confidence that you're strong inside. After all, your source of happiness lies inside. Because it's not taking anything away from anybody else, you don't have to be afraid of them.

Yuttadhammo Bhikkhu also has a nice YouTube video talk on the Dhammapada verses on the broken gong.