There has been a lot of discussion about this sad topic in other threads. Perhaps when you're feeling more steady you could do more research. I think it's inevitable that at some time in everyone's chicken-keeping experience they'll have to face this issue. That's why I advise all chickeneers to consider their options & resources right from the start, along with setting up brooder boxes & adjusting their heat lamps.
The thing with chickens is that their owners have to learn to diagnose and treat most, if not all, of their ailments & injuries. There are some of us who have professional care that is available & affordable. But most of us must learn to care for their needs ourselves.
Which is but one of the many reasons for the popularity of this forum.
Possumqueen said it best, what's easiest for the bird is often the hardest for the owner to do. I try to keep in mind that my ailing bird is depending on me to end its misery quickly & effectively, and that gives me courage to do it. If it were out in the wild some predator would certainly attack it in its weakened state, and the bird's last moments wouldn't be very nice.
I use cervical dislocation (breaking the neck) to dispatch my ailing birds. And like Karen71 said, there is some reflexive flapping & other movement afterwards, but I know once the neck is broken the bird is truly dead & feels no more misery. It's okay to put them under a bucket or blanket & walk away until they're still.
You can hold the bird's neck down on the ground under a broomstick held firm with your feet and give a good yank up on the bird's legs, you'll feel the neck bones crack & the head goes limp. That's when their wings start to flap & it can be alarming, thinking the bird is still alive & trying to fly away. It's not, it's just reflexes.
I will hold the bird under one arm, grab the head with the other hand, and yank out, then bend the head up. Then I'll let the head hang limp and continue to hold my bird so she doesn't flap around.
I bury my birds at the drip line of flower bushes or trees. That way their bodies help the plant to grow and in that way, they continue to live.
Where do you live? Maybe there's a BYC buddy you can find nearby to help you. I would if I were close.