In my opinion, you should not apply for another credit card at this time, but the reasons have nothing to do with your credit score.
First, while you are paying off old credit card debt you should not, in my opinion, be charging new purchases to a credit card. I wrote another answer about this recently. Use cash, checks, and a debit card to make purchases at this time and learn how to only spend money that you already have in the bank.
Second, despite the fact that you have been getting credit card offers in the mail, if your credit score is poor right now there is no guarantee that you would get approved for them. And the ones that you would get approved for might not be the best cards.
Third, your credit score will improve as you pay down your old debt and you continue to make on-time payments.
Once your debt has been paid off in full, you can evaluate your situation in a new light. You will have been functioning without using a credit card for some time and can make a better decision about whether or not you want to start using the credit card you have, get a new credit card, or just continue life without the use of a credit card for a while.