wouldn't I just be able to translate my American prawo jazdy into Polish?
No. You need a specific document called an IDP - and as you already know, only the AAA can issue one for an American licence.
There's no other way, unless you choose to exchange your US licence for a Polish one and pass the theory test. Even then, it isn't a certainty that they will exchange it - it depends on whether your licence conforms to certain international standards. Poland is rather strict on this.
Contact the local WORD for more details.