I would book either directly with WDW, or with a Disney Specialized TA. It is Disney who sets payment policy, whether you book with them, AAA or any TA. For packages it's $200 deposit, and payment in full 45 days before arrival.
Disney does release discounts, and Disney Specialized TA agencies know the minute they are released, and are on the phone getting you the best price.
I book myself or sometimes go with a Specialized TA. I find it very easy to work with them, I like to make monthly payments, whether its a room only or package. All I have to do is send off an email to the TA and it's done.
I always book my own flights, and ADRs, regardless of how I book.
AAA is a general Travel Agency, and isn't actively watching Disney to know when discounts are released. You would have to do your own work, then phone them to ask for them to apply discount. IMO this takes away the perk of going thru a TA if you have to do the work anyway.
There are some exceptions, and some AAA TAs are very good with Disney, but they seem to be the minority.
If you are new to WDW and booking, I highly recommend using a Disney Specialized agent. They have a lot if knowledge, and will do the Disney Math to figure out the best price & promo for you, can advise you on the resorts, dining, park plans, etc.