Should I take up another balance transfer card?

Honestly speaking, I don't see any other option right now. Shift the balance to another balance transfer card. You can close the present account. It won't drop your FICO score drastically since its payment history is quite short. Make sure you pay off the balance on the next balance transfer card soon. Otherwise, you would never get out of debt. It's kinda vicious cycle.

Applying for several different cards with low introductory rates may negatively affect your credit score. But right now you don't have any other choices.
If you still have difficulties to pay off your credit card balances, you can opt for a debt settlement program.

Yes, get another balance transfer card to transfer the balance. Try to pay off the balance as soon as possible. Cut down the extra cost to save money. Reduce buying unnecessary things using your credit card. Save money aggressively to get out of this debt.