We are 'alive' according to the scientific definition of life, and a rock is not. There is no arguing that.
Existence is something completely different, it's a human formed concept and is more flexible. In my mind existence depends on being sensed.
When a person dies maybe some chemical reactions are still taking place, however these involve the decomposition of the body and will be driven by other bacteria etc. The bodies biochemical pathways will no longer be functioning, we will no longer be 'alive'. Also our senses will no longer be function- we will be unable to see, hear, feel, smell, taste. Because of our inability to sense the environment around us and ourselves we will also cease to exist. Our body still exists- other living beings can see the body, feel it etc. But that life that used to be in the body has ceased to exist. A rock, like a dead body, exists, but it can only be sensed, it has no ability to sense- it exists physically but not emotionally or sensually - it is not alive.
Obviously you can look at existence through many other lenses (religions, philosophies) and get different answers. There is no way of knowing whether one idea is right.