A person I will always remember is my grandmother, because she was an amazing person I have ever known. ==> either "the most amazing person I have ever known" or simply "an amazing person."
My grandmother was a person everyone wanted to be around, she always put a smile on our face without even trying. ==> Everyone wanted to be around her, and she always put a smile on our faces without even trying.
She had the funniest laugh and always was telling jokes,
which was probably
why we were always laughing!
My grandma would put anyone before her no matter what the situation was,
even if it was out of her way she would be there to help us.
She was very caring and always had a welcoming home for us; my cousins and I always liked staying with her on the weekends. ==> this sentence is fine for the most part.
My grandma was like my best friend I could talk to her about anything, and she knew exactly how to solve my problems. ==> also fine.
She also made the best turkey dressing and banana pudding, I always looked forward to Thanksgiving walking into her house smelling all the warm food she had waiting for us. ==> this should be two separate sentences.
I will always cherish all the memories I had with my grandma she raised me to be a better person and to look at things in a positive way. ==> also two separate sentences
She was a strong, loving, caring, funny woman and had many things to give to others,
that everyone that met her will cherish forever
I'm not sure how this fits into your application holistically, but overall, I think the content is a bit weak. You talk about how great your grandma is, but that doesn't really say much about who you are as an individual. Also something to consider: My English teacher's stance on writing about relatives is that it can be difficult to do well just because anyone who reads it will have a knee-jerk reaction to compare them to his or her own relatives.
Just my input.