Anyone can have social anxiety. Social Anxiety Disorder is another story, but only a qualified doctor can give you their medical diagnosis. For the purpose of Y!A, you can certainly identify with social anxiety if you want to.
If you think parts of your story are lame and cringey, I would recommend taking a break and editing it later. Then you can touch it up. Ultimately, I want to remind you that no one's writing is perfect and no one will ever reach the point where they don't have any weaknesses. I'm sure if you posted your classmates' writing up here, we could all find plenty of lame and cringey parts pretty easily. It sounds like you've been work-shopping everyone's stories, so I'm also sure everyone was given guidance on something. Some kind of advice. Some kind of "Well I liked this, BUT..."
We had a writing club at my college and there were few instances when there were no suggestions for improvement. I believe it only happened a couple times with some short poems. With stories, there's a ton of room for imperfection. It's also noteworthy to mention that sometimes people disagreed on what needed improvement. Some people might have thought a line didn't fit while others adamantly wanted it kept in (besides the writer).
I remember we had some people who didn't want to read their own work aloud. I read for someone a couple times. The club president had a good reading voice and always volunteered to read for people. We had fun with it. Maybe there's someone with a "reader voice" in your class. You could ask your teacher about having someone else read your work. If not, practice reading it aloud to yourself a couple times. It'll help you feel prepared as you read it.