It sounds like you are doing well already. Paying $6300 a month should pay off your debt in less than a year.
You have no late payments and good interest rates, so just keep it up and wait it out. Just stop making big charges on your cards.
Might be a good idea to consolidate to a personal loan. Question: are you going to continue procedures? Is that why you want to apply for new cards?care credit may be a good option...
Like webhopper said, you should consider opening a personal loan. Otherwise, opening a Chase Slate or any other card that's offering 0% on balance transfers for at least a year will be a good alternative solution.
If you can take out a personal loan for a lower APR that would be a good option. You didn't mention limits but if they are close to maxed you FICO score is getting hammered and that will be a big impediment to obtaining a signature loan. Many credit unions offer debt consolidation loans that may be a bit more lenient with the maxed out cards but otherwise clean credit. With a local CU you can also talk face to face with a LO and explain the situation. Your is a lot different than someone who just lived beyond their means. Good luck!
If zero % consolidation is not an option, I would list them smallest balance down to highest (regardless of interest). Pay minimum on all but the first on the list, hammering it out with avengence. Once it is paid off, carry that former minimum balance and tack it onto the next one in line. Hammer that out with avengence and tack that former minimum onto the next. and so on and so forth. Otherwise known as a debt snowball. This method also doubles as a financial behavior modification rather than interest mathmatics.
“Beware of little expenses. A small leak will sink a great ship” – Benjamin Franklin
Gardening since 3-26-15
I was in a similiar situation, less debt, but I also net much less than you. I didn't have to consolidate, although that may be a proper option for you. Give yourself a realistic goal and live by this mantra: NUKE MY CREDIT CARDS. Start with the highest APR and work down the list. Know that it will be a bit a journey and it will take a lot of discipline.