To answer your question, if you kill a deer in January, you CAN legally still put in for the fall hunts, ostensibly to get a bonus point. Should you draw a fall deer tag, you may not kill another deer as this will exceed your bag limit for the calendar year.
Also, a deer hit with your vehicle and claimed on a salvage permit does NOT count against your bag limit. I hope MoreD was joking.
But in reality the fact that you have two tags isn't really something that's tracked as there is no requirement to report deer kills except in the areas where there is a mandatory check in station such as Kaibab or some special hunts like 3-bar used to be. I've been asked to show my unused archery tag before while on a draw hunt so it is not something that you want to abuse and run the risk of losing privelages for a period.