Congrats on the new job! What a great opportunity. I hope you don't encounter too many angry customers. If it's the kind of store where people bring their cats and dogs, maybe they are more likely to be friendly?

I have the same problem when I have to confront people, but only when someone is more than mildly angry. I turn (and see) red, shake, and then sometimes feel "off" for days (and want to quit my job). Working in customer service has actually helped me to get over this a lot. What I do in the moment is go through the angry-customer-diffusing steps I learned:

1) Apologize (for something, even if you can't change anything)
2) Repeat back the facts of their complaint/problem (this makes them feel heard and makes sure you're addressing the right issue)
3) Empathize (If they're emotional, they're probably a little embarrassed even if they don't seem like it; this makes angry people angrier. If you can express the fact that their response is reasonable/understandable, they will be less defensive and calm down.)
4) Ask what they want (sometimes it's smaller than you think; also, this gets them focused on the solution instead of venting their problems at you)
5) Offer a solution (tell them what you're able to do in positive terms and let them accept or reject it)

I've found that if I focus on doing each step well, I'm less likely to focus on the other person's scariness. I still feel the physical upheaval, but it's smaller and I can keep talking. The steps themselves often help to diffuse the situation and solve the problem. If not, at least I know I approached the situation in a professional way, so I'm less likely to ruminate about it after.

I've also learned that if the worst happens and I do burst into tears or get red and sweaty, that I'll live. No one likes to be yelled at. Lots of perfectly well adjusted people get their feelings hurt at work and break down. It happens to most people at least once. It doesn't make you unprofessional or weird (which is what my brain is telling me).

Over time, I've been able to deal with people better because I've had practice. I've also learned over time that the bad customers are a very small fraction, and the nice ones make up for them. I am confident that you will be able to deal with it better and better as you go. Do you have someone you can roleplay with ahead of time?

I second lonegael's suggestion about CBT. There are some good workbooks on anxiety if you can't do therapy right now. Breathing exercises and relaxation techniques are helpful, too. Good luck to you in the new job!