Lovely to get your email. My answer to your question is: YES.
YES go to grad school.
YES look for work/internships.
YES do whichever comes first/feels best.
Here are the caveats...
All the schools you mentioned are the best, which you should definitely aim for and which would be lucky to have you. Some of them are very expensive, however. If there's any advice I can give you, it's this: please please please don't go into insane debt to get a graduate degree. If you're lucky to have a scholarship or parental support then it's a little easier to recommend grad school. The life of a screenwriter can be extremely volatile in terms of financial stability, it's good to not pin yourself under a mountain of debt if you can help it. You'll want to keep the overhead low when you start making stuff, which leads to--
Yes, keep looking for work and internships like the one you had at GMM, of course, always do that. But more than anything MAKE YOUR OWN WORK. That's your ticket. Write something, find people who you like and trust, and figure out how to beg/borrow/steal (which is really the work of good producing) and make it. Then do it again. Get better at it. Find your voice. Get better at your voice. Make something else. Find better people to work with. Trust them. In so doing you'll learn to trust yourself more. Keep at that process and sooner or later you'll make something that will get you work. And that work gets you more work, and ideally you keep that ball in the air as long as possible. It'll fall time and again, but you get better at picking it up. It's a wild ride.
You can do the above in grad school or not in grad school. If you need structure, grad school is very helpful. If you are radically self-motivated, you might not need grad school. If you need a couple years (or 10) to travel the world, or take a few odd jobs (I worked as a bartender in Times Square for years) to see life, scrape the underbelly, meet weird people, find the zeitgeist that makes you want to fly out of bed in the morning, then take time to do just that. (I believe Werner Herzog said it's better to be a bouncer at a strip club if you want to be a filmmaker than to go to grad school. I don't recommend that line of work for you, and I believe grad school has more merits than that, but his sentiment is dead on).
If you ever need a letter of rec, I'd be happy to write one.
Can't wait to see what you do out there!