One CC has zero interest but will expire in February and unlikely will pay off the balance by then.

Check the terms on this. Sometimes these have clauses that it's not 0% interest but deferred interest. That means if it's not paid off you will have to back pay all the interest that's been accumulating, not just interest going forward.

Would I be best to put all of these into a balance transfer card or focus on just paying one off at a time?

Depends on the exact terms, sometimes there's a balance transfer fee, but you'd probably want to run the numbers to see if it nets out better. If there's a card that has 0% for a bit and a low transfer fee that could work out for you.

Should I continue putting anything in savings as well or just focus on paying off the debt? What's the best way to navigate this?

I'd personally argue that you should pretty much cash out your savings and pay off your CC debt now. However that's risky if you lose your job you have no cushion. Still $4500 on CCs is going to very easily balloon into a bigger problem, so at a minimum stop saving and put everything into your debt.