You seem to have it backwards - the Oxford comma is present when the comma between the penultimate and ultimate items in a list is /present/, not when it is not!
"This book is dedicated to my parents, Ayn Rand, and God." <- Oxford comma
"This book is dedicated to my parents, Ayn Rand and God." <- no Oxford comma.
This is a humorous example advocates of the Oxford comma use; the second sentence can sound like the author is saying her parents /are/ Ayn Rand and God - if read when expecting the Oxford comma.
If you are using the Oxford comma, "I jumped, I hit my head and I cried" explicitly indicates that the crying occurred with or because of the hitting your head. If you are not using the Oxford comma, it might have been unrelated in time or causally.
Using it or not is a matter of style, and people have VERY STRONG OPINIONS about it, but neither is more "correct". What /is/ important is that you are consistent - either use it or don't, everywhere. You can't switch back and forth - or you'll end up doing something silly like the above example eventually.
(If you are writing for a company, journal, or in a class - you may be told explicitly by a style guide or supervisor/teacher to either use or not use it. So, in that case, do whichever is required!)