Hello from another immigrant from Asia!
I'm sorry school has been tough on you. Like you, I am not comfortable in social situations and I usually don't know what to do to join a group or make friends (I usually wait for others to make the first move). We speak English as our primary language in my homeland and I still found socializing here hard. I can't even imagine how it is for you when you have to socialize in a language you're not as comfortable in. I'm glad you've reached out today.
Yuki, this may sound like scary advice, but the best thing for you to do is to actually keep speaking in English even if you feel self-conscious. The more you speak, the more you practice the language. Conversely if you stay quiet because you feel you're ruining the vibe, you're denying yourself the opportunity to learn how to fit in.
Start talking more to the people who ask you to be their groupmate. They're not just asking you out of kindness, they do like you enough to want you to be on their group. Put yourself in their shoes - if you met an international student in your homeland, you'd ask them to be in your group because you wanted to know them, right? Not because you disliked them.
You're putting a lot of pressure on yourself to be perfect in English when you're still getting used to speaking it as a primary language here. Because of that, you're very self-conscious. The result is that your mind goes blank when you feel called on to respond or socialize (which themselves are stressful situations for you). So be kind to yourself and give yourself time to adjust. Practice speaking to yourself in English in front of a mirror and trust in your speaking abilities because a lot of your hesitation probably comes from a lack of confidence. One more thing I've found useful when getting used to speaking another language is to try and think in that language instead of translating from your main language into the other language. It takes more time and effort (and it's more difficult), but you'll see the difference because you start responding more quickly.
It'll get better soon. Don't lost hope.
Feel free to reach out if you want to.