Very helpful staff walked with us to the Etihad counter to see if they by any chance would be there, but no. Then we walked to Terminal 1 (lucky the airport is not that big!) to see if they would be there, but no... Argh.
They could not let us into the country on tourist visas, as we were already in the system as residents, so for a while we were a bit at a standstill. As it turned out, there had been some miscommunication and the visas were still at the Etihad HQ in town.
The solution, after a little bit more than an hour of this walking back and forth between offices and terminals, was to leave our passports with the Immigration, and come back the following day with the visas after having picked them up from HQ. I wasn't all that happy surrendering our passports, but what to do?
In the end, it all worked out and we had arrived in our new country.
One taxi ride later, and we were also in our new home. Hello Abu Dhabi!