If he really has a voracious appetite and has lost weight, I would bet on hyperthyroidism. Are his nails really long, like looking overgrown and maybe starting to approach his paw pads? Is his appearance becoming unkempt, like his coat has little mats that he didn’t used to have?
Drinking for a long time at the water bowl and vomiting, I would bet on kidney disease and/or failure. He possibly has gone long enough without vet care to be far into kidney failure, which also causes nausea (which usually causes anorexia, unless too hyperthyroid to ignore sensitive tummy because sooooo hungry) and vomiting if also hyperthyroid.
Hyperthyroidism also masks kidney disease - they often occur together or within a short timeframe of each other - and are common in older cats.
Go to the vet, get a Senior Feline Panel done, which should include a CBC, Chem 25, T4, and urinalysis with culture and susceptibility. If your vet suspects kidney failure and/or hyperthyroidism from any of that, then there should be at least a “free T4” and a renal panel added to the above (unless the initial values are so high the verifications aren’t needed).
Personally, cats I have who are over 10 years old get senior panels done yearly. My cats can’t tell me when they feel a little icky. The panels tell me before the cats can if their kidneys or thyroid are starting to go. Much like people, the organs start to fail before they feel bad, and once they feel really bad, so bad that their people notice, they’re pretty darn sick.
If you get any results from the panels that he’s having a problem, making a euthanasia decision won’t be so hard. Whatever you do, take it from a shelter worker, euthanizing him will do him a far bigger favor than taking him back to the shelter. If you take him back to the shelter it will be a death sentence for sure, and not a nice one that he could have with you. The stress of going back to a shelter and dealing with all the changes in daily life, along with health issues that may or may not be diagnosed/treated properly by the shelter, all older cats do when they get returned to my shelter (which is one of the best shelters to get returned to, with a full-time clinic to take care of the cats) is basically get hospice care until they die. There’s little hope of getting re-adopted for cats over 10, especially if they have health or behavior problems.
If you have to make a decision between return to shelter and euthanasia, please pick the latter.