I could feel the spinal cord was severed, but they did not die for a minute or longer.
Yes they did. The flapping is a reflex reaction, completely normal, that happens with chickens after death. It does not mean the bird is still alive.
I was concerned because the bones are snapped, or cervical is dislocated, but they are still getting blood and oxygen flow to the brain.
If the spinal cord is severed, then the bird is clinically dead. It doesn't matter if there's residual blood still going to the head for a second or two after. Consciousness and the feeling of pain are both rooted in the brain, so if the brain is dead, then it cannot feel pain and the bird cannot be considered alive. Don't beat yourself up - if the spine was severed, then the birds were dead. The rest is just reflexes.
What I use for my chickens is a large pair of wire cutters. The kind with long, sturdy blades. They cut the head right off, no ambiguity. Pruning shears/loppers can work, too, but you have to get the right kind because not all of them are sturdy enough to do a clean job. It was funny, I went into Home Depot once to ask them about their sturdiest loppers. I had to explain what they were needed for... The staff just about fainted

Then some lady said hold on let me get somebody... She comes back with this old, seasoned Haitian gentleman from another department, and we had a nice chat about our respective personal history with farms and butchering. Turns out he grew up with farm animals, and knew just what I needed. I love my wire cutters, they do a great job, especially in combination with the killing cone. Make sure you start cutting from the back of the neck, not from the throat. That way the first thing you sever is the backbone, which is what matters. The throat is secondary. Sometimes I can't cut all the way around the whole neck in one go, especially on larger birds, but as long as the spine is severed, I rest easy knowing that the animal feels nothing, even if the head is still dangling by some skin.