Your English is really fine compared to most people I've talked to, so don't doubt yourself too much. To be honest, English isn't my mother tongue. In primary school. I was at a disadvantage when I communicate with people as my parents couldn't speak English. Hence I have to slowly learn it. Learning has no shortcuts, sadly.

I would suggest you to improve your English is by reading English novels, reading English newspapers (For difficult words, Uncle Google is there to help you.) and most importantly, communicate in English. You may be afraid to talk to new classmates because they do not want to be friends with you but it isn't true though. If you are someone nice and helpful, eventually you will get along with your new classmates. They should have similar interests with you, so use that opportunity to make some early friends.

If you need a grammar checker, go search for an application called Grammarly. This one is good for you to correct your grammar mistakes and yes, it is helpful in assignments. Just the free version will do, you don't have to go for the premium one.

In fact, I will be moving to US in mid-August to pursue my studies, so technically I am facing the same trouble as you: Finding and making new friends.

Just message me in GT if you need more tips, I will be willing to help out.