It makes sense in that this is exactly what I would have done in the past, keeping on swapping from one 0% to another, but it catches up with you in the end. It does your credit rating no good, either (I learned only recently) to keep moving the money from one card to another - you never actually address paying off very much of the money you've borrowed, and the lenders have no way of knowing that you're not going to spend up to the max on all your cards each month.
You'd be better off getting a bank loan from your bank to pay off the £600, and borrowing a bit extra to pay for your partner's Christmas presents.
We stopped buying each other Christmas presents when we couldn't afford it - we were only adding to our credit debt to do so, it just wasn't worth it.
You will also find that at some point you will be rejected for a new card, if you keep getting new ones just for the zeroes - it's not worth it.