These past issues will be heavily scrutinized when you apply but are not necessarily deal breakers. However, I would suggest you wait a longer period of time. Five to ten years of sobriety plus having fully dealt with the depression/anxiety will look far better.
Of course, the best way to find out is to approach larger departments and speak with their recruiters about your past. They will appreciate you being up front with them and generally give very straightforward and honest answers to your questions.
Police Officer, Washington State
B.A. Law and Justice