Is "masteral program" the phrasing that is used? Not master's program? (I've come to learn that my American ears often question what is perfectly acceptable elsewhere, but this is a new one for me. It's a doctoral program, but I've only ever heard master's, not masteral.)

I think your letter is very good in tone.

work was so hectic that (not work had been too hectic that)

Did you drop only one class? "I had no option but to drop Economics" or "I had no option but to drop Information Systems" but not "drop my Economics subject."

Regarding the missed deadline, I'd be skeptical that your schedule left you with no option but to miss the deadline. I would suggest you put a period after dropped subject, and then say something like "I greatly regret that while I was focused so intently on my new duties, I missed the deadline for requesting a leave of absence."

"the end of the first quarter of this year" is a rather long way of saying "since March"- is there a reason you can't say "Since March..."?

confident that I should is a little contradictory. Are you confident? Confident that I will be able to.  

Finally, I look forward rather than I am looking forward.

Good luck! (And a belated congratulations on your promotion.)