In the earlier years, I had to pay the departure tax at the airport. AA used to collect it at a separate area as you moved up the line to check in. Then, AA began to include it in their ticket price.
I did ask AA if the tax was included, they told me one time that it wasn't and when I got to the counter I did not have to pay.
So, asking the airlines when you buy your ticket, isn't necessarily the correct answer.
I always tuck away what ever the departure tax will be next to my passport. Then, if I don't owe it - I spend it at the duty free shops.