No Disney is not lying to you, and calling back will not help fix the problem. However, you can book the DDP with an AAA discount.
You need to call an AAA office directly and speak to a travel agent there. Through an AAA agent at an AAA office, you can book a package which will include hotel, park tickets, and DDP. You can have an AAA discount applied to the room portion of the package, generally from 10-20% off the room part only, depending on the resort class and season you're going. You want to verify with the agent that you're getting the AAA discount rate, though, because if the AAA rate is sold out for your resort on the dates you're going, you won't necessarily get that AAA rate.
However, if you call Disney directly, you will only be able to book a room only (that means no tickets and no DDP) with the AAA discount.
Do note, however, that you cannot double up the discounts. So, if
free dining is being offered, you could not get that plus the AAA rate at the same time. Still, if there are no other discounts available, AAA is a good deal. We saved over $300 last year off 8 nights at the Poly (with tickets and DDP) and this year, we're saving closer to $400 for the roughly the same stuff, all in July when there are not other discounts available.