I know that exact sick feeling in your stomach you're talking about. I do feel like you should take care of yourself first and foremost. It wouldn't hurt to look into other places to see if they have hours that work better for you. If you can't find anything I would think that taking a day to go see a psychiatrist is acceptable because if you don't take care of your mental health you won't be able to properly focus at work the way your employer needs you to.