Lots of good advice here already!
My suggestion is to tackle this from the language side first. It's really frustrating to be "trapped" behind a foreign language, feeling that you can't express yourself. It gets harder to learn languages as one gets older, and learning a language is like climbing a staircase: you struggle for what seems for ever with something, then you jump up a level, and everything seems a bit easier. Until you realize you're on a plateau, and you don't seem to be getting better. Again. And again. And again.
So that's one thing: even when you feel like you're standing still, as long as you're immersed in a foreign language, you are improving. Even if it doesn't feel like it.
Best advice here is just to find low-stress arenas to practice. Join a sports club, chess club, book club, film club, martial arts club... you get the idea -- find some people that share your interest and hang out. Talk without having to deal with a foreign and difficult subject at the same time.
That might help with the anxiety connected to English.
You mention English is your third language, are you using both the other two actively? If you can, it is usually much easier to learn a language if you can avoid switching back and forth -- and just stay in the new(est) one all the time. Talk, think, do everything in English. Avoid phone/video calls in your native tongue (within reason) -- try to stick to email/letters. If you think it's hard, try to make a deal with anyone you call regularly, say you'll try to give it three months. Let your brain work for you.
Don't worry about the syntax errors in code. Just code more, and you'll make fewer mistakes. Try to code for fun, or at least outside of a class setting, without any pressure. Maybe sign up at codeeval.com and try your hand at some of the easy exercises first.
Many of the problems there are pretty much lifted from project euler -- and that's a good resource too. But with codeeval everything is wrapped up nice, very little setup needed, and you can just play around.
Last, but not least -- you mention you're attending college -- try and talk with one of your TAs, professors or fellow students. See if someone might have the time to pair program with you, on something not directly related to class (again, try to avoid to stack different kinds of pressure anxiety). For example, you could make a deal to try and solve some easy codeeval (no, I'm not affiliated, it's just an easy source of a few exercises) challenges.
Finally: the best way to learn is to make mistakes. It's certainly much better to make mistakes than do nothing. And if you make errors while pair programming, not only do you learn from your mistake, your partner does to!