This is a plea.
Please, please don't wait to talk to your veterinarian about a behavioral problem until you are so upset about it the only thing you want to do is relinquish your pet.
There are so many resources we have in today's medical world and also in our understanding of behavior. Inappropriate house soiling in cats is a disease that can be managed and in some cases simply cured.
Kicking the can down the road and thinking a rescue is going to take a cat who has a house soiling problem is unrealistic. It's cruel to the cat and it's difficult for the veterinarian. I have actually been in situations where I or a colleague have been asked to euthanize a patient for a problem that could have been treated if the owner had asked for help years or even just months ago. Trying to "fix" the problem by dumping your cat with a behavioral problem on an already overcrowded shelter is sentencing the cat to death. I'm sorry I'm being blunt- no- I'm not sorry. People need to know there isn't an island where abandoned pets go. No one (or VERY, VERY FEW special people) are going to adopt a cat from a shelter where "reason for relinquishment" is cited as house soiling.
Sometimes, the willingness to add an extra litter box, to change the litter more frequently or limit the cat's access to certain areas are the only things that need to be done. There is kitty "prozac" that can be used in more severe cases, but most of the time I don't even need to resort to that. Most of the time it just involves a conversation. Sometimes, in difficult cases, you can even consider making the cat an outdoor cat. If that's what it's going to take to keep the cat from being dumped, the risk of a cat outdoors if it's properly vaccinated and cared for is a lot better than a cat with a death sentence.
Please- whether it's a dog or cat, call up your veterinarian and talk, or schedule an appointment. We have no problem troubleshooting these situations with owners. We do have a problem with people having a callous view of pets as property.
In some extreme situations, we do exhaust all options and these are special situations where maybe that cat has a severe anxiety disorder or mental illness, but these cases are really rare, but when both the vet and owner know they did literally everything they could, everyone can sleep ok at night. Asking a vet to sentence your cat to death because you don't want to add a litterbox? I will restrain from my vitriol on that one.