If you're getting paperwork, then it looks like you're in. But, I don't blame you for being cautious. You're doing the smart thing.
I think a lot of people underestimate how difficult the AI process can be. You've waiting a long time, so once you get your initiation date, you'll be very relieved.
Don't get discouraged if you don't hear anything this summer. That's normal. I bet things will get hopping for you really fast in the fall.
I'm excited for you!