I agree, giving her to someone else (unless she is to become an outside only cat) won't work. The most common reason cats are abused is because of things like urinating outside of the box. But at the same time, I think it may be a little early to jump to euthanasia. Working for a vet, I'd say probably the most common reason for a cat to suddenly start urinating outside of the box is a urinary tract infection. So it's very important to see the vet before you try to do any sort of behavioral modification because if she's doing it for a medical reason absolutely nothing you do is going to stop the behavior until the medical problem is addressed and treated. Most cats that are urinating outside of the box because of a UTI will go right back to using the box again once the infection is under control. However, some cats will develop an aversion to the box or the litter or the litter box location after an infection because of the pain they experienced during the infection.
I have one cat, Smokey, who has had frequent urinary issues all his adult life. He is on a prescription diet because otherwise he was getting 2-4 UTI's a year on regular cat food. When he has an infection, he will urinate on towels, piles of laundry, or blankets on the floor. At one point, he did develop a litter box aversion and would not use the box to pee at all. We ended up retraining him to the box by confining him to a large dog crate and using a tricyclide antidepressant for a period of several months. It was getting to the point where between us being angry at him all the time for peeing on something else and him being locked in a cage that we were seriously considering his quality of life and euthanasia was on the table. Thankfully, he is now off of medication and faithful about using the box again, except when he has an infection. Which he hasn't had in over two years (and he's been good about peeing in the box for 4 years now)!
I have another cat, Kisa, spayed female, who will occasionally urinate outside of the box for behavioral reasons and has never had an infection. For a while she was doing it mostly because the litter boxes are in the basement and she was getting shut upstairs (so that the baby wouldn't fall down the basement steps) with a full bladder. Mostly she was peeing in things that resembled boxes, like a shoe box lid or something like that. Then it became a behavioral thing, she was doing it because she was angry about the baby even though she had access to the litter boxes. Shutting her in the basement first thing in the morning seemed to help a lot. Now she does it if the litterboxes are not clean enough, and cleaning the box is all it takes to correct it.
Good luck with Pixie. If she does not have a medical reason for the inappropriate elimination, then you will have to start the long process of figuring out the behavioral cause and addressing it. You may have to provide new boxes in different locations, try covered and uncovered, and try a few different kinds of litter to see if she has a preference. While you are trying to get the behavior under control, and until you can get to the vet on Friday, it would be a good idea to confine her so that she has limited access to your furniture and the things she is currently peeing on. If after trying to correct the behavior you are unable to get her to use the box, then it would be best to either make her an outside only cat or have her put down. But I would give it a few months before turning to euthanasia. It's a long, frustrating process but so worth it in the end if you can get them back to using the box.
