Certainly not a semicolon in the second example.  Semicolons separate sentences, and since 'jeopardizing. . .' is a clause, that won't do.  
'I simply wanted to disappear, as I felt I had betrayed the people I admired most; jeopardizing their respect and trust for me' is simply a run-on sentence.  It's actually three sentences smooshed into one.  I think better is:
'I simply wanted to disappear.  I felt I had betrayed the people I admired the most and jeopardized their respect and trust for me."
Another comment is that 'simply' is usually a garbage word.  "I wanted to disappear" is much stronger.
"It was a home away from home and by my senior year I was friends with nearly every teacher and administrator. " This is a cliche.  There's always something better than a cliche
However, it was not till my senior year that I began to fully appreciate their dedication and guidance.  Use 'until' instead of 'till.'  'Fully' is a garbage word.
During that year I cheated on a computer science project.  'During' is obvious and implied, so not really needed
It's been several years and I still vividly remember my parent's and teacher's disappointed reactions.  It's should be expanded as it's really 'it has.'  Since tense is being emphasized (i.e. this is a past event), using the past indicator here will help.  Shy away from adverbs when an adjective can be used.
Yet, it was also very revealing as I barely spoke a word.  However is better than 'yet.'  Also don't use 'as' as a conjunction too frequently.  It's a bit awkward and can be used to make things interesting, but it's going to be distracting if you use it often
how she was still so proud.  Don't need 'so' here
This was followed by every teacher on the review committee  Stronger is 'Each teacher on the review committee followed,' 
They knew the effect they had on me and I hope one day I too will have a similar influence on my students, where expectations are held in the highest of regards. Run on sentence and confused tense (you suddenly jump to the present with 'hope')
It was those teachers, my second family, who helped Stronger is 'Those teachers, my second family, helped.'
access to such a quality education.  Such doesn't add to the sentence here
Such an education would not only prepare someone to follow their dreams, but also guides A (strong) eduction prepares someone to follow their dreams and also guides
Personally, I want my talents and experiences to contribute to the greater good and education is the impetus through which I find the strength to sacrifice and serve others. Probably better as two strong sentences
It is my sincere desire  You probably already know what I think about 'sincere.'  We'd like to believe all your desires are sincere
However, I am cognizant how difficult Esoteric words can break the flow of an essay.  A simpler word like 'Aware' would probably be better
Although eager, I am not quite ready to be in a classroom and succeed to my expectations. Urgh.  A bunch of things here. . .  Avoid clauses before the subject.  Lead with 'I am' instead of 'Although eager.'  'Quite' is another one of those garbage words.  'Succeed to my expectations' is awkward and 'meet my expectations' is more accepted.
receive more than adequate preparation. Probably a stronger positive than 'more than adequate,' which is pretty weak, could be used here
transition to Arizona, allowing myself Better would be 'and allow me'
As well as perform research pertaining to how best to attract more students, especially women and minorities, to the sciences. Sentence fragment, as already pointed out.  Could be 'I also wish to perform. . .'
Luckily, I have had many Luck really wasn't involved.  This word can be removed
At my high-school, during the day,   You don't need to mention this was during the day unless you then say what you did during the evening
at a local inner city school. There I prepared  Actually, here a conjunction would be suitable.  'inner city school where I. . .'
In actuality,  Ugh
This had a profound impact upon whom?
Since then I have mainly tutored mathematics to college Stronger is 'I have continued to tutor mathematics with . . '
While teaching friends is always a challenge, I have Try 'Teaching friends is always a challenge, and I have'
There my most difficult 'There' is unnecessary
: improving my patience and .  It has improved my patient, 
Most of all, I've come to appreciate how every day I can learn from others, as much as they can from me. Most of all is not needed.  The end is a fragment.  You can use a colon there if you want
In all honesty, I would happily teach any subject under most any circumstances I would be happy teaching any subject under any circumstances
Yet, I find Do not need the 'yet'
I will forever enjoy 'Always' is more appropriate than 'forever' here
passion in pertinent ways to  'Pertinent ways' is very much implied here, so it can be dropped
Yet, I know  Another unneeded 'yet'
that all subjects have; hence, I am greatly  . . . of all subjects.  I am greatly . . .
forward to endear in my students an  to endearing my students with
Yet, above Yet again, another yet
Please don't take this as harsh.  I do a lot of editing and proofing in my line of work.