How do I speak with my professor about my mental health?
A number of the things you mentioned you've been thinking about the situation, i.e. you're taking advantage of the teacher's kindness, how your reputation will be ruined etc possibly fall under the category of unhelpful thinking styles or cognitive distortions which are common misguided ways in which people think that often leads to an unhelpful outcome. People with depression and anxiety can be particularly bad at thinking in these misguided ways. 

I would recommend seeing a professional mental health expert like a GP or Psychologist as soon as possible and not waiting one month for the councillor. There are lots of tools that professionals can teach to help manage stressful situations like these. Seeing a professional doesn't imply that there is anything wrong with you either. From your post, it seems like you're having a natural reaction to an incredibly high workload and that you haven't gone completely 'off the rails' so to speak. Seeing a professional can help prevent things from getting any worse.

In the meantime, I think the action you should take becomes clearer if you acknowledge that your previous thoughts might be the result of unhelpful thinking patterns. For example: You don't actually know that your reputation is going to be ruined. That impression is most likely the result of some assumptions about someone's reactions. In truth, there are a number of possible outcomes and it is futile to worry about the one possible negative outcome as if that outcome is certain.

I think that the best thing for you to do is to sit down and write out everything you need to do, how long roughly it will take to do and when you are going to do it. If you really need an extension, go and ask the teacher for one and explain the situation and the action you are taking to fix it. 

You might find worksheets such as these useful to reduce anxiety levels about the perceived consequences of actions.