First there are 7 different kinds of a FFL- a dealer is ONE of them- HOWEVER- No, you do not need to have a Federal Firearms License to PURCHASE a Class III firearm. Matter of fact, you can't GET on 01 FFL unless you can show you intend to ENGAGE IN THE BUSINESS of buying and selling guns. Can you carry it? Depends on your state law- the AOW you describe is a handgun. Check w/ your state licensing agency. You buy one like any other gun, EXCEPT- you must have it transferred to you by a dealer that has a FFL, with a Class III endorsement- not all dealers do. You will need to apply to the BATFE for approval of the transfer. You wil need to submit a fingerprint card, and approval from your local chief law enforcement officer, along with payment of the transfer tax ($200 for a full auto, SBS, suppressor, etc- $5 for an AOW) . Bottom line- find a Class III dealer in your home state, and he will guide you. Oh- and be prepared to wait.
What we have here is... failure- to communicate.