Here are a few thoughts for you:
July 3, 2005. I had dived headfirst into 'no man's land'. A mistake? Only time could tell.
Generally speaking, in the U.S. we only use single quotation marks when there is quoted material within another quote. We will put words in quotations if we are using that word ironically.
The journey is still vivid in my head and I can still recall it like no time had passed.
You use the present tense here and then then past tense. You use the word "still" twice in this sentence. If you can get rid of as many of the "to be" verbs as you can, the essay will be stronger. Stronger verbs will also take down your word count--taking out the fluff and leaving more room for the meat. Here's one way of rewording: The journey resides vividly in my mind as if no time has passed. Nineteen words down to thirteen.
What I couldn't quite remember was the reason my parents had for venturing steadfast into the vastness of America's 'Last Frontier.
This sentence is a little clunky. Again, pare it down to its essence. There's a stray apostrophe here ... you might have meant it as something to be inclosed in single quotation marks, but you don't need the punctuation in this case. Here's a shorter version: I couldn't quite recall the reason my parents ventured into the vastness of America's Last Frontier. Twenty-one words down to sixteen.
My family and I had boarded a cruise ship at Vancouver Canada set due on Alaska.
We already know that you went to Alaska so you don't need to repeat it here (not that it is wrong, but a denser essay will read better). Put a comma between the city and the country. "Set due" is a bit awkward. Here's a quickie revision: My family boarded an Alaskan-bound cruise ship at Vancouver, Canada. Sixteen words down to eleven and no more passive voice.