Reenlistment ?

Ive met a couple people that did this but i will say its uncommon. Also units dont want to pay for you to go to airborne. They want to pay for other schools that would benefit their already airborne soldiers. its more of a pain in the ass for the unit since you arent qualified to take part in regular unit airborne operations till you get it. I would just make sure airborne is in myorders. This is the only sure way to know you will get to an airborne unit. Also, chances are if you are airborne you will go to bragg. And if by chance you don't go with your first orders you will go on your next set of orders. a large portion of people in bragg try to leave bragg. And i can tell you from experience its kinda hard.

You may reenlist for airborne school. You will attend airborne school TDY enroute to your next unit. You can’t reenlist for Bragg because there are no non airborne 11B authorizations there. Only HRC can waive the airborne requirement and they don’t do that for 11B
When you reenlist for airborne you will find out what your follow on assignment is 4-6 months prior to your course start date.

Yes, it is true that you can do that.
Talk to branch about it and tell them what you have in mind. They might be able to get you into airborne and your choice of station. That's what I did and got both for reenlisting. Now, when I asked he did caution me bragg wasn't available for assignment (it was full for me) but that was a year ago almost.