I had a similiar problem just yesterday. I have automatic payments set up on my credit card. On the first of the month it takes money out of my bank to make the payment. Unknown to me, my minimum payment went up and the amount taken out was lower then my payment should have been. This resulted in a $30 late fee and my APR going up to an insane 27.9%!! I called them and asked for it to be lower. The lady on the phone said, "I'm sorry, we don't have a better rate to offer you at this time. Make your payment for the next 6-12 months on time and call us back then." I told her there was no way that I was waiting that long with such an unreasonable APR and asked to talk to her supervisor. She placed me on hold, and then a minute later came back and said, "My supervisor adjusted your rate to 15%."
There is the key: When you call, you are talking to idiots that don't want to be bothered helping you out, and probably can't anyway. Go above them to get anything done. Now that you have a different card tell the supervisor that if they can't beat the APR on your other card that you are just going to cancel the account because of the way they treated you.