Do I need to complete my pre-req science courses before applying?

Are you a Texas resident or applying in Texas? They have 2 year Biology requirement (12 +2). TMDSAS also provides a deficiency report after reviewing your application showing what requirements are not being met by your transcript. If you are in your final year and apply early enough, you can plan some additional classes for the final year to fulfill those requirements (i.e., outstanding ones).

Unfortunately, AAMC does not do that since every school can impose their own requirements.