I agree with Nan 99%. Only if you're the King of Great Britain, Queensland, and Northern Territories (or P.L.'s sovereign, the King of Southeast Belgium) should you retain the Roman numerals after your predecessors croak.
Many Americans are a bit slow to realize that, but at least we're smart enough to avoid royalty in the first place. (Britain's is only held together by the powerful Matching Hat/Purse/Coat Purveyors lobby. Wise up people! Trade the whole sorry lot for one good orthodonist!)
But I'm not sure about the III-to-Jr bit. My recollection is that it should be III-to-II, which at least acknowledges that there might have been someone bearing your name before your Dad.
I used to be a "II", but coming from a family of dirt-poor illegal alien Scottish coal miners (who despised the Royals and anyone else with money), it seemed unbearably stuffy. I didn't use it much.