Having trouble responding to professors' emails due to anxiety, I have crippling anxiety around "authority figures", and I'm irrationally deathly afraid to the point where I will cry in a room when alone with them, likely due to my upbringing, namely, always being afraid of my verbally abusive, but very loving and well-intentioned, father.  And so I tend to leave very important and very kind emails from professors unanswered.  This is unprofessional of me, and so I was wondering what some of you here have successfully tried doing to counter this level of anxiety.

The authority figure can be a professor, a fireman / police officer looking to help me, my basketball coach.  I've experienced similar with all of these people.  I am sometimes just mute.

I see a doctor and take meds already, but I am looking for other ways to cope.

Is there a cool, fun strategy to try to implement, say, respond to emails within 5 minutes of reading them, even if my response is incomplete?  Aim for a one-sentence reply?  

I don't know, I'm just thinking, but I know I need a lot of help.
You are doing the right thing seeking some medical advice. But the way to become more comfortable in this, as in most things, is:

Practice. Practice. Practice. 

The first steps are painful and you may fall. Eventually you learn to walk and probably to run. 

My doctoral program took a few years longer than it might have because I was too afraid to speak up on my own behalf at a certain point and got stuck (really stuck) with an unhelpful advisor. 

Later, I took on the issue head on and taught myself to look people in the eye and to speak up. It was very difficult at first. Now I do it without thought or worry. 

It is just a skill. Learn to apply it. Haltingly at first, but it is extremely unlikely that bad things will happen if you speak up. If people don't know your needs they will not be able to assist your academic growth. Just. Do. It. Grow into the person you want to be. 

Sometimes you need to just pretend to be the person you want to be.