Wow, I feel your pain! I think most of us who have had careers in customer service management have gone through similar situations - the transition between being a "doer" and a "manager." The sign of a mature senior management is that they would recognize the challenges of this transition, and help you prepare for it. To set you up to succeed, as it were. On the other hand, an inexperienced senior management would just throw a person into the deep end of the pool, and expect them to figure out how to swim. So what I am saying here is, don't beat yourself up.
I think the previous two posts have some good advice, and I would like to throw in a little more. Businesses succeed or fail based on their customers - the cusotmers' perception of the company's product or service, and the customer's perception of how well they are treated. Therefore, it is imperative that the people who serve the customers be well-trained, capable, friendly, courteous, and articulate.
It sounds like, in your previous role as an account manager, you excelled at these things. Your senior management was hoping that you could magically impart these skills to your customer service team. This, as both previous posts mention, is a common assumption of senior management everywhere. Just because a person can DO something well does not mean they can teach others to do it well, let alone manage the performance of others, without specific training on how this is done.
I was thrown into a similar situation about 20 years ago, and was lucky enough to have been given a book called "Managing By Influence" by Ken Schatz. A year or two later, I was lucky enough to attend a training program delivered by Mr. Schatz that fully explained and utilized the concepts in the book. I can honestly say that these things changed my life, as a manager. It taught me how to delegate, how to handle internal matters between employees, how to get my boss to give me the resources I needed to get the job done. This book is out of print, but is still available in used book stores, and occasionally on amazon.com. It's worth seeking out.
One more comment for you, about the angry customers. This is a topic near and dear to my heart, as I have written extensively about them, and positioned my consulting practive around them. Yes, angry customers are hard to deal with, and can make you feel like crap. HOWEVER... they are great if you are trying to figure out how to improve your business and / or department. Think of it this way... small, trivial issues rarely make customer ANGRY. If a customer is that mad, it has to be about something important. To paint with broad strokes here, I would say that the anger is about one of two things. 1) Your company (or team) dropped the ball on some service or product issue. This is where training or product improvement will remedy the situation, and make you a stronger business. 2) The customer has misunderstood what he / she was supposed to get, and even though your company / team has delivered as they promised, something in the company's sales or marketing communication led that customer to believe he/she was going to get something more, or different. This is where you try to get the customer to help you identify where the message got garbled and the confusion was born.
Best of luck to you, and hang in there!!!
Angry Customer Strategist