I am also taking prescription medication as ordered. I was able to obtain a Secret clearance in 1997 with no problems. I have never did drugs, committed a felony, declared bankruptcy or have ever betrayed my country in any way. But, all kidding aside, it would seem quite unfair for you to be denied the ability to serve based on your having a well researched chronic illness for which you are receiving responsible preventative treatment. I would hope that they would not hold this against you.
Depression is nothing to be ashamed about.
Depression is a treatable illness. It is not a 'weakness' any more than having a chronic health problem such as diabetes is a weakness. Phobias are learned fears that are greater than is warranted by the inherent dangerousness of the feared thing. Some people are afraid of public speaking, others are afraid of closed spaces, and some are afraid of driving cars. You don't say how bad your accidents were, but I can imagine that they may have been disturbing enough to convince you that driving is a very dangerous thing to do.