Euthanasia of pets

So we've had our siamese cat for about 14 years. He's always had accidents outside the litter box but now they are increasing as he gets older. Two years ago he chewed/licked all the fur on his belly and arms off. I took him to the vet and there were no illnesses or anything wrong with him. He was hairless about a year and then he suddenly stopped licking. He has his fur now and no more licking issues but now he defecates all over our house. He pees all over and doesn't use the box. He used to use it most of the time, except if someone forgot to scoop it a day. Now my mom is making me take him to the shelter or to the vet to have him put down. I don't like either option and I feel like euthanasia is unfair. What would you guys do? Yes to euthanasia? No? Find him a new home? Everyone is telling her to put him down but she's making me do it and he's been my best friend for many years, I feel like I'm betraying him..he's just getting weirder ):

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