English is one of those classes where it really depends on your professor or your TAs. From my experience, not being a native speaker will not earn you much sympathy since there are so many of them.
I took American literature and my teacher didn't care much about grammar or spelling. He looked more at literary aspects of my writing and how much I paid attention in class. I did all the readings religiously and actively participated in discussions. I am not a native speaker myself. I came here for college but still got the highest grade in class.
Ask your TA for help, go to office hours and ask people to proofread your paper!
Best of luck
I would suggest practicing the grammar all the time so you get a better grasp of it more quickly. Like while on the internet, do not use internet speak, speak grammatically correct to the best of your ability, and ask your friends to do the same.