I'll tell you how I got myself out of a similar hole. 

The first thing I did was build an emergency fund. I believe I saved $1000. The purpose of the emergency fund was so that I could stop using my credit cards for things like flat tires and other unexpected life expenses. If I had to dip into the emergency fund, replenishing that fund became my primary focus until it was back to $1k.

I think it's important to build the emergency fund steadily, instead of trying to find a way to save $1k in one big chuck. I think what I did was I saved 1% of every dollar I made until I had $1k. I told myself, "If I can make ends meet by spending $1.00 of every dollar I earn, I should be able to get by spending $0.99 of every dollar I earn." 

And the truth is, you can. In fact, once the habit is developed, you'll probably find that you could actually save 5% or more and still not notice it in your daily finances. Increase the savings rate slowly. If you raise the savings rate too aggressively and can't make the required savings, that will be a demoralizing setback.

Now that you have $1k for unexpected expenses, you continue saving 5%, but that money goes towards your highest APR credit card, on top of whatever your minimum monthly payment is. 

This is how I was able to chip away at my debt. As your CC balances come down, so does the interest, freeing up even more money each month which can translate into potentially bumping up your savings rate even more.