I get very depressed too. Am now actually. Some things though, especially when I am depressed, I simply ask myself "what else have I to lose?"
If the answer is "nothing" and it is not something that needlessly endangers me or someone else - I do it.
Here - you are already hopeless. Should you get denied .. it will seem to validate your hopelessness but effect nothing else - unless you think it would tip you over the edge into suicide.
So - if it is only hopelessness holding you back .. then you have "nothing else to lose" and thus, should apply. If it would effect anything else negatively, you should not.
You know all your circumstances and all your emotions, not me - so that's the best advice I can give.
Probably is not what you want though..
I do hope whatever you do includes being good to yourself. Take care.