I'm going to give you some peek into how this would work inside the admission office
If you apply to programs that conduct personal interviews, English language is one of the most important skills they want to access. Those programs are Baruch, CMU, UCB, Princeton, etc
For those programs, if you have great profile but low AWA as the only flaw in your application, your chance of getting an interview is very good.
Now the second set of programs that don't conduct interviews (Columbia, NYU, the rest), they will use everything in your application to best gauge your English skill.
You don't study in an English speaking institution, which means you will have to take the TOEFL and do well. 100-110 is par and you should try for 110+.
If you spend several years in the US doing undergrad or graduate study, that experience would work in your favor. You don't need TOEFL and besides no language test can supplement the experience living among English native speakers.
Your essay would play some part in showing your ability. If you have a low AWA and have an essay that reads like Mark Twain, it's a red flag. If you have a high AWA but your essay that reads like it belongs to a 8th grader in a public school, it's also a big no-no.
The point is, you can't spend your life speaking another language to improve your English. You have participate in an environment where English is the main language.
The internet is a good place to start. Do you have a blog where you write English daily? Do you participate on forums like Quantnet frequently? How often you argue your point with native English speakers?
I would want to make sure that if I admit you, you would be able to step off the airplane and talking to American without missing a beat. Why you ask? Because you would be seeking internship right at the end of your first semester in the US and it would look bad for the program if the interviewers can't understand what you say or write.
There are a lot of way that you can improve and "prove" your language skills. Don't expect people to use your AWA and TOEFL alone to be able to judge your English. It's really hard.
Keep in mind English is a small part of your whole package, you can easily have great scores and still fail to do homework in other part of your application. But that's another question.
As a personal general guide, AWA 4-4.5 would be in the middle of the pack. Your AWA is way off that range. If you are confident in your English, retake it and get 5-6.
Disclosure: I'm doing admission for one of the programs above that does not interview.
