Auriga - I think I can relate to how you feel, at least partially. Throughout my life thinking of suicide has always been with me, although mostly as a kind of escape exit "If it gets worse, I can always kill myself". And not for any philosophical reasons, but simply because life sucks. Last week it got so bad, I even started thinking about technical details, fortunately I managed to climb out of my black hole.
What keeps me alive is mostly feeling of responsibility, I do not want to let down all those people who trust me, my principal, my students. Which is strange, as I also suffer from impostor syndrome, so whenever my professional achievements are praised I feel fear that people would find out that in reality I'm a cheat.
I've been in therapy, it helped some, not with everything. Last week I seriously considered going to a psychiatrist but did not get to arranging an appointment. Maybe I will, one day, or maybe I will go back to therapy. Or maybe my life circumstances will change enought to make me feel better.
Anyway, exercise is OK, endorphines make a great drug, but I would not stick to running. For me it's the most pointless activity ever, so I do aerobics, weight training and dancing instead. And it helps. Mostly. I also have people to whom I can talk, although the majority of them are on the internet. People around me, my colleagues, my students, know nothing about my real life, one of my colleagues even calls me "Sunshine". It seems I'm a master of disguise.
If I were you, I would keep going to psychiatrist. Sometimes the medication takes some time to kick in - or possibly it's time to think about trying something new? Often it's not the first drug that helps.