I would also contact the broker on this one. The broker does have some responsibility on this problem you have with the company they brokered it to. They should be able to supply you a copy of the contract between them and the transporter that did the work which states them (the transporter) will take care of any claims and that AAA is not resposible for damage done during the transport.
I have done work for AAA Automovers and they are just a broker, not a transporter.
That is a problem with alot of the companies that you contact, they act like they are a transporter but they are only a broker. Once they have your deposit ranging from $50.00-$250.00 they then put your car out for someone else to transport.
I have gotten screwed by 2 brokers in the past. One I had to wait 8 weeks for $100.00 check for a job I did. I had also asked for more $$$$ like $150.00 and they said there wasn't anymore to give. Come to find out they charged the customer $350.00 for the transport, allowing them to make $250.00 broker fee and making me wait for my $100.00.
Also, sometimes you book your car with a broker and it never gets picked up, 2-3 weeks go by and still nothing. You call them up very upset and they say "well, if we bump it up a hunderd or two, that will help it get moved quicker". The reason for that is say they quoted you $1000.00 for a move from Northern MI - South FL. They took your credit card for the deposit amount of $250.00 and the remaining balance is paid COD $750.00. Now, not too many transporters are going to do an enclosed transport for $750.00 that is 1500 miles and with the liability of a $160,000 car. I mean yes, you can make $$$ if you are using a 6 car hauler, but first you would have to be in Northern MI for it to be worth wild, and there isn't too much that goes up there enclosed. Once in FL, there is alot more there so that isn't a problem. I think you get the idea.
Please let us know the complete outcome with your situation and also Good Luck!