Please don't surrender your sick, elderly dog to a shelter. His quality of life will decrease there. The likelihood of him being adopted is extremely slim; what you will do is subject him to a confusing, stressful situation before he's euthanized. Assuming you've had him since he was a puppy, 13 years is a long time to have a stable home, only to suddenly be abandoned. Your shih tzu (and you) will likely cope better if an at-home euthanasia is performed for your cocker spaniel.