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You cannot humanely euthanize your bird at home.
You need to see the vet to either diagnose the bird's problem and provide treatment or to seek euthanasia.
Let me know if you still have questions.
Your local animal shelter or humane society can help you to euthanize the bird at a minimal cost. Give them a call. I'm sure it can be done much cheaper.
There should be no problem euthanizing the bird if the animal shelter still uses a gas euthenasia chamber. Not all shelters still have these chambers so you should call around.
The problem with birds is that many euthenasia technicians are not familiar with the methods necessary.
Using a gas chamber would eliminate the need for a specially trained person.
I hope this helps.