The only problem for some people carrying a high debt burden is that the lower rates are usually reserved for people with solid credit records. Before shifting your credit card account to another company, call your current credit card issuer. Explain that you are thinking of moving your account to take advantage of a competitor's cheaper interest rate, lower annual fee, or better overall value. Your current company probably doesn't want to you as a customer and may match the offer. It's well worth the phone call.
Many credit cards are what the industry calls variable rate.
In other words, the interest rate you pay is pegged to an underlying interest rate, typically the prime rate. The rate usually changes based on some formula. Fixed rate credit cards do not change as often as variable rate cards. But the fixed interest on a credit card is not the same as a fixed rate mortgage. The interest rate on the latter never changes.