Grammar usage

Some people would use a comma after I'm sorry, but most punctuation guides say it is incorrect.  A period or a semicolon is the correct punctuation.  The case for using a comma here, other than sloppiness or ignorance, is that I'm sorry is intended to be an interjection at the beginning of the sentence, in which case a comma would be correct; however, I tend to side with the stricter rules that insist that, no, it is a full clause (subject, I, plus verb am plus adjective sorry), which makes a comma incorrect. 

If you get rid of the subject and the verb, and just use Sorry, then a comma is okay because it is definitely an interjection then: Sorry, I can't....

Another option is to fix the sentence with the conjunction but, in which case a comma is correct: I'm sorry, but I can't....

There is an additional punctuation error in your sentence: you need to join the final two clauses with a semicolon, period or a colon.  I got into an argument on Grammarly recently about which is better, but I stick with my opinion that a colon is best, because the final clause explains the reason for the middle clause.

I’m sorry, but I can’t go to the party tonight: I’m snowed under with homework.
 

As for blizzard and snow flurries, they are both winter weather-related nouns that I think you can readily find in any dictionary.

I hope this helps.

Shawn