Is “marking” something only done be female cats? We have two male cats and have not such problems. (Now about scratching everything they can get their claws into...)
My adult daughter was broken hearted about a cat she got from the shelter, had it neutered, had it in a one bedroom apartment in Chicago and it marked all around the entry hallway. She tried everything for about a year and a half l, a cat psychologist, changing the brand of litter, having many litter boxes, changing the food, giving it lots of exercise, but it was marking her hallway as a way of owning the territory (others walked their dogs or cats in the hallway.) It was a marvelous cat in other ways and very bright. He would play fetch with you for hours at a time if you let him and he wasn’t a scratcher. In much desperation, she took it to her vet who had her sign over the cat to her and the vet had it as an office cat and he did really fine there. Everyone is really animal lovers in our family so it wasn[t lack of love. Sometimes you get cats that do that and with my daughter’s cat, the older he got, the worse it got.
I would try and return the cat or talk to the vet about the cat and see what he or she suggests. It is very unpleasant to have that oder around especially with a new house and kids. Maybe the cat could be a mouser in someone’s barn. With our daughter’s cat, it was just too confined an environment for that particular cat.
Cordio
Usually the other way around. Unaltered males are the usual culprits, although some altered ones will do it also. Haven't known a female doing it, but guess that could happen also,
We’ve had male cats to do this—they’ve all been neutered too and some of them sprayed all their lives.