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I have to agree here 100% One of my cats, the one we've had the longest and the very first pet my husband and I ever got "together" (2 years before we were married and before we were actually engaged even), has had frequent urinary problems. He's very prone to urine crystals, which cause him to urinate outside of the box. Even on c/d prescription food for the type of crystals he has he would still get UTI's about twice a year (we have since switched to a different prescription food and he hasn't had a UTI in over 2 years) At one point, his urinary indiscretions became behavioral. We had to retrain him to use the litter box. It took about two months to get him back to using the litter box. During that time, we were giving him transdermal amitriptyline (an antidepressant in humans, is sometimes used to help treat behavioral urinary incontinence in cats) and he was segregated to our GSD's old airline crate with his own litter box. At first, even in such a confined space he didn't use the box. I was cleaning urine off the floor of the crate twice a day. Slowly he became better about using the box and we were able to wean him off the amitriptyline and back into the regular household. Fortunately, he has never had a relapse and all of his urinary indiscretions since then have been triggered by a UTI.
The vet I work for has a cat who peed on the car seat right after they brought their daughter home. It was very clearly the cat being unhappy about the baby! Once he adjusted to the fact that this new little person was going to be living with them he went back to using the litter box.
I unfortunately also have to agree that if he really really won't use a litter box, the shelter would not be a great option. He may be adopted out if he uses his box at the shelter, but if he starts urinating around his new home he's going to be right back at the shelter again. And most shelters will only adopt an animal out so many times before they euthanize because of obvious behaviors that make them unsuitable pets for most adopters.
If you can afford it, the invisible fence might be the perfect solution for you to keep your cat outdoors 100% of the time and out of the neighbor's yard. But you will want to go with the Invisible Fence brand, not a cheaper electronic fencing system because the cheaper systems are less adjustable and generally not suited to cats. Also, the branded fence includes several training sessions with a professional to set it up properly and get your cat used to the fence so that it is not a traumatic experience.