Is it permissible to take out an interest-bearing loan to pay off an interest-bearing credit card debt, or is that considered paying off unlawful with unlawful?

You must repent to Allah and get rid of the interest-bearing card through lawful means, not by entering into another interest-bearing contract, because what you mentioned does not constitute a necessity that permits you to do so. Look for lawful means, such as Tawarruq, to obtain money and pay off the card debt. Increase your supplications, especially the prophetic supplication: "O Allah, I seek refuge in You from anxiety and sorrow, and I seek refuge in You from inability and laziness, and I seek refuge in You from cowardice and stinginess, and I seek refuge in You from the burden of debt and the oppression of men."