RTA governs driving license rules in Dubai now, which is why some things seem to have changed (the rules probably haven't, but the RTA is more strict about implementing them). Other emirates still via traffic police.
Usually driving licence comes from same emirate as your visa. Licence valid for all UAE but each Emirate issues their own. Cars supposed to be registered in same emirate as visa.
Exceptions if you live and work in different emirates (show accommodation contract to prove where you live). In the past that could work your advantage (speaking from experience), but the RTA can be so bloody-minded at times, I wouldn't be surprised if it meant you couldn't get a licence at all, nor register a car.
From
Dubai driving license info.
Re the OPs question, I'd say just go to Abu Dhabi police and try. I think you'd have more luck than if you were dealing with the RTA. Take some proof that you live in Abu Dhabi (telephone bill?).