The best thing to do, the professional way to handle the issue would be to give two week's notice. From the way you phrased your post, I understand that you get this but you feel incapable of doing so.
Perhaps the most honest thing to do is to request an immediate medical leave because of your issues (whatever they are). Then after getting a few days of perspective maybe you would be better able to face either giving 2-weeks notice or quitting immediately.
It won't be the end of your life, and I can also promise that it won't be the worst thing that either of your managers has had to deal with -- these things happen and sometimes there is nothing to be done. But I DO think that you should find a friend or confidant who can be supportive and help you work through this.