Generally, for most people, it's not recommend to get into churning if you are in CC debt. Mostly becuase no one wants to advise you to create more debt that you might not pay off and depending on the amount and your score, etc you may have trouble getting approved for cards.
However, I broke the rule. I do have old debt, 3 cards actually at pretty low rates thankfully and I decided to give it a go. I have not had an issue getting approvals. I have actually raised my score and I've been able to make a big dent in paying off my old cards. I never put anything on my new cards that can't be paid for in cash immediately. I broke a second rule by making my husband an AU on a card and in doing so I was able to get his score up (helped his debt to income ratio) which will help a lot when we finally purchase a house, hopefully this year. Our old debt was actually due to a house that turned into a money pit and we trusted our realtor when we probably shouldn't have, long story. Churning has allowed me to concentrate on the old cards, finally travel a bit and feel much less stressed. I have learned a lot and I'm better able to guide my children. I can't say it will be the same for everyone though and I take full responsibility for the decisions I make regarding CCs. I've had approvals for Southwest Plus and Premier, CSR, CSP, Ink Preferred and SPG Biz.
So, churning while in CC debt isn't advised, but if you are diligent and stay on top of it, it can be done. It sounds like you have a plan and I think you won't have an issue with approvals based on the CC debt. I'm not advising you to proceed, just sharing my story so you can make your decision.
Now I'm going to go hang my head and hope I'm not banned and labeled a rule breaker