I think this is too much about everyone else in your family, and not enough about you directly. The PS should be almost entirely about you.
Now for the writing. I'll give you my feedback on your first paragraph. If you generalize the advice to the rest of your writing, I'll take a closer look at the whole thing on your next submission.
“Let’s get started with an easy question,” the alumni representative said pleasantly, “tell me about your family.” I was at my first college admissions interview at the local Starbucks. I smiled nervously as I shifted uncomfortably in my seat. What was supposed to be a casual question to make me comfortable made me instantly uneasy. Whether I was at a job interview, chatting with a new acquaintance, or making small talk with a stranger, this question never failed to make me uncomfortable. My family’s atypical, complicated structure made this question difficult to answer, often leading me to give vague, ambiguous answers, hoping for a nod of indifferent approval rather than risking judgment from giving a full, accurate description.
Adverbs are evil. Don't use them unless you have to. We can see he's being pleasant, you don't have to tell us. "Alumni representative" should just be "alumnus/a". "Representative" does nothing extra to inform the reader. Also, he didn't ask you a question. And, "get started" should just be "start." Try to cut out any unnecessary words or phrases. How about:
“Let’s start with something easy,” the alumnus said. “Tell me about your family.”
Your third sentence has some of the same issues. How about:
I smiled and shifted in my seat.
You want the reader to see that you are a bit nervous. You want to avoid having to tell the reader directly what's going on in your mind. This is what people mean by, "show don't tell."
You should cut down on the adjectives. Adjectives are often unnecessary fluff. Other times they are a crutch. Instead of showing, you end up telling. In your last sentence you doubled up on nearly all the adjectives. And it did nothing for you. How about this for your last sentence:
My family’s complicated structure made this question difficult to answer, often leading me to give vague answers, hoping for a nod of indifferent approval rather than risking judgment by giving an accurate description.
The meaning is exactly the same, but now there's less clutter.
This sentence is also on bit on the long side. You could break it up like this:
My family’s complicated structure made this question difficult to answer. I'd often give vague answers, hoping for a nod of indifferent approval rather than risking judgment by giving a more accurate description.
I think you're also under the impression that you're not allowed to use contractions. That's just silly. Without contractions your writing sounds stuffy and too formal. It doesn't read the way people speak. Say these out loud:
"I don't want any more pizza."
"I do not want any more pizza."
The latter sounds robotic. It's too formal, or maybe even aggressive.