A few things. You use the word that
a lot. Seriously.
Get away from the emotionalism. You were upset, yes. But they don't want to hear that. You also need to cut the fat from this letter while, at the same time, owning up to what you did/didn't do.
I am writing to appeal my academic dismissal from xxxxxxxx. I was upset to recently receive a letter informing me that I am subject to academic dismissal. I am ashamed to admit that my low GPA is a reflection of my effort during my spring 2016 semester, but I take full responsibility for my actions.
then I wrote:
I am writing to appeal my academic dismissal from xxxxxxxx. I am ashamed to admit my low GPA is a reflection of a lack of effort during the spring 2016 semester. I take full responsibility for my actions.
In particular, do you notice how I eliminated the word but
? Your usage of it changes the character of the sentence. The term 'but' implies a sentence's direction is changing, but (see!) your sentence did not. Splitting it into two simple sentences and eliminating the incorrect conjunction helps take care of that.
I also want to point out, further in your letter, you mention 'I will have a weekly schedule of activities while making sure my academics come first.'
That just looks, to them, like you are going to keep on partying on, hoping you'll somehow make it. Instead, you need to show them you've got a plan to succeed. You do not need to cloister yourself, but at the absolute minimum you need to let them know you are getting a tutor, you are going to study x hours/week, and you are going to use a calendar and follow it in order to assure you study for quizzes and hand in assignments on time. Right now, it still appears as if you have a cavalier attitude. Maybe you do; maybe you don't. But (there's that word again; see how it shows a reservation in the mind of the writer?) what you have written here does not show me seriousness of purpose, and I get the feeling the committee won't see it, either.