I think this sentence is a case where "that" rather than "this" (dog) would be a better translation.
This would mean "Nie mamy jeszcze tego psa" = we were promised we're getting a dog, but the dog hasn't arrived yet. But here, most probably the dog that we had was lost/sold/it died.
"tego psa" is Genitive (here) or Accusative for "this dog", not "a/the dog".
"the" is considered a possible interpretation for "this", because that's often what you in fact mean by saying "the dog". But "a" is wrong, it's a specific dog.
Oh, I missed this question. Well, almost half a year later you certainly know that, but anyway: "Nie mamy już psa".