Tell them they need to be quiet.

It's a good thing to go out of your comfort zone once in a while. Hard for me, too. But, it makes you stronger. Baby steps.

Someone recently posted something to the effect of "Why does everyone try to encourage introverts to be more social, but they never encourage extroverts to be more introspective and quiet?

Being social is seen as "normal" and anyone who isn't social is often seen as weird or broken. That couldn't be further from the truth. Everyone is different and extroverts seem to forget that. We all have different needs. Some people need to shower every day, some can get away with every other day. Some people rely on caffeine every morning, some don't. Some people need lots of contact with other people, some not so much.

I wish I knew the perfect thing to say. Too often I am seen as stuck up or shy when I am around groups of people who are doing something that I don't want to participate in. For example, I was at dinner at my sister in law's family's house who I did not know that well. There were 12 or so people who were playing a game. I was happy watching them from a distance, but when they practically begged me to play, I shut down. I had to go in the bathroom to get away for a while.

My (not) favorite thing that extroverts say when you decline is, "Oh, come on, we don't bite!" That's not the point at all. My only suggestion is to politely decline. Maybe be upfront if you have to. Tell them that you have a limit on social interaction and that you've had enough for today. They might not like it, but sometimes you have to do what you have to do.

You could tell them they should spend more time alone, and think about what each individual’s needs are.

They like to socialize and want the same for you... They just don't understand that you don't like to socialize... They are people persons... You are not....

I would simply tell them that everyone has different needs and they are welcome to speak to their own, but should not assume their needs are universal.

Because they don't understand. If someone tells me to socialize I either clam up, go sit in the corner and ignore everybody or just leave.

Isn't that what your doing here ?

Smile, wave and say... No.

It's extremely annoying!