How do cat shelters (or other people with multiple cats) handle it when two cats
don't get along?

Some of you may remember my previous posts regarding the saga of Old Cat and New
Cat, two cats who just can't seem to get along. For those who don't, Old Cat is
an 11-year-old neutered male. Up until 2 years ago, he had a dog to play with;
she passed away from cancer. For a while after the dog died, Old Cat seemed to
miss having another animal as a companion; he would wander around the house
mewing, as if searching for his lost buddy.

So we went to the shelter and adopted New Cat, a spayed female who was about 6
months old when we adopted her. I've lived in multiple cat households in the
past, and never had any problems introducing two cats to each other. In my
experience, they always became fast friends. I figured if Old Cat would be
buddies with a dog, then another cat would be even better.

I was wrong.

Old Cat and New Cat don't get along very well. New Cat seems to stress Old Cat
out - he runs away from her whenever she approaches. In the beginning there was
a lot of hissing and batting of paws, but that seems to have settled down. New
Cat is very playful, and likes to chase Old Cat. Old Cat responds by running
away and then, when no one is looking, he craps on the carpet(!!!) In almost 10
years prior to adopting New Cat, Old Cat has never even once failed to use his
litter box; now he'll do it every time we let him out of his "sanctuary." New
Cat, on the other hand, does not seem to be genuionely aggressive; it seems to
me she only wants to play. Unfortunately, Old Cat just isn't into the whole
playing thing anymore.

It's been a year and a half since then, and unless we want to keep picking up
cat feces from the carpet we have to keep Old Cat locked up in our bedroom while
New Cat has the run of the house (or vice-versa).

So I'm wondering what people who have multiple cats do in a situation like this.
Do you just rip out all the carpet, put linoleum on the floor, and let the cats
work things out for themselves? :)

We don't want to give either one of them away. Anyone have any
industrial-strength solutions for us to try?

Thanks for your help!