Exhaust every option counseling-wise and legal-wise, but NO level of discharge--even "dishonorable"--is so worth avoiding to put your life in mortal risk of such a horrible decision.
BTW, alcohol may not be the root cause, but it doesn't help and feeds the problem. Vicious spiral and all that.
WHATEVER you do, don't take the coward's way out. You and your family both deserve better.
Badger, your use of the word "coward" is a disgusting insult. However it turned out, I put myself our there for my country. It's so funny that we are all heroes but civilians turn on us in the blink of an eye.
I'm really sorry to read all this -- I wish you the absolute best. Depression and anxiety can be debilitating, and I understand that contemplating suicide just comes with the territory. The important thing is to realize that you have an issue and to seek the necessary help.
I don't know anything about how a discharge would work, but I'd say there should be enough help and support available through the proper channels in the Navy. My understanding was that they were much more lax about discharges for such things than they used to be.
Definitely get help -- speaking from experience, it makes a difference. Let me know if you need anything.