So basically the final date to pay your fee is on the census date.
Yes. You'll effectively be a citizen March 21, with 10 days to spare until the census date rolls over. The.very.next.day. go into the uni and get it all sorted. Do it in person, not over the phone. Good news that you got a quick ceremony, some people have to wait a few months!
I think the liability fee that you're talking about is the student amenities fee (SA fee)? Mine popped up last week, saying I owed UNSW $163 odd, but there's SA-HELP, which is just another loan help situation like HECS. Apply for it the same day you get your HECS sorted. You should've received an email about it last week, but go here to read up and be prepared for when you do fill it out in a months time.
https://my.unsw.edu.au/student/fees/SAHelp.html