Are there specific areas he's always going to aside from right beside the box like always going to a certain corner in the living room etc? I'm wondering if the previous residents had cats that peed in the house/apartment - perhaps your cat is covering their smell with his? Usually this would happen within the first week of moving in but it's possible.
As Kristin7 mentions an enzymatic cleaner will remove the smell, don't use ammonia based cleaners - to a cat they smell smiliar to pee & usually will go again to cover that smell.
I would say a health check at the vet is a very good idea, stress can cause many different things - my senior kitty developed HyperThyroidism after we moved.
