She said it's one of her headache morning , therefore there's nothing I should be worried about.
Since it was one of, morning should be plural:
morningsAs I tell him my mother's condition and our address, my hands were shaking and soon my whole body was shaking.
"him" is vague, instead, try to make it a specific person such as someone at the hospital/ etc
The ambulance came with the shattering sound of siren and carried my mother on a stretcher.
The ambulance carried your mother? Do you mean someone in the ambulance car such as a nurse?
I rode the front seat
You rode "in" the front seat
I knew something was really wrong and I should be holding on to myself since my mother needed me the most in that moment.
The first impression I got when i read "I should be holding on to myself" was an actual physical move of like hugging yourself. If you're trying to communicate that you should be taking control of yourself or in control, you might want to rephrase it. This is just an opinion though, you don't have to change it if you feel your first wording was right.
I didn't want to be as a burden but as a person who she can trust on.
You can omit both "as"'s & I think who could be whom. You can also omit "on"
Overall, good essay and nice reflection at the end, I think you answered the prompt very well and was able to demonstrate clearly how this specific moment in your life matured you :)