DIY euthanasia is tricky, because to do it in a humane way usually requires access to chemicals and knowledge of techniques that the average pet owner doesn't possess. It doesn't help that the internet actually has some suggestions out there that are 100% fucking inhumane. Maybe living on a farm you'll have things like scales and experience with animal handling and injection though. You might be able to get your farm vet to help you out with this process.

Your humane options:

Find another vet

shoot it with a .22

inject it with (at a minimum) 100mg per kg of weight with pentobarbital via intraperitoneal injection - mind you that's just one page, I'd do lots more research and practice injecting some fruit or something before trying it on a squirming animal.

smash the head in with a hammer

just wait until it stops eating and drinking, they tend to die pretty fast then. In the meantime, if you're worried that it might be in pain you can give it some painkillers, like ibuprofen - (15 mg per kg of hamster weight, mix it in some apple sauce or something once a day).

And please don't adopt any more animals if you're not willing or capable of providing them with adequate health care. Hamsters are just as worthy of compassion and vet care as any other animal.