Every single card 3rd party processor I've used (as merchant) automatically provides name and address (including IP, etc.)
Same here; usual business practice is, if it's customer data, you save it with the transaction. I always did, but not full card numbers as this had greater legal requirements, but definitely card holder name. So if you used a CC on AW, then they will probably have this on their database too (but not exposed to the outside obviously). You will need to hope AW never get hacked. One benefit of this is that they've surely assessed their risk from lessons learnt so far from AshMad, including inside jobs?
One solution, like the OP, is to remove your AW profile; after a month it is meant to be non-recoverable. I would not be 100% sure that this removes all your data from their database. It may just be a delete 'flag' with the data remaining on the database for ever. This is what AshMad did, even after customers paid for deletion. It would be darn useful, for all those who are anxious, if this could be found out?
If you have or haven't ever used a CC then the lesson from AshMad is, don't, ever. Only use adult sites that take cash vouchers. UKash, which many have mentioned, is now (taken over by) PaySafeCard. Used one hassle free on a betting site recently and AW take these.