You are contractually responsible for any debt that is in your name, even if someone else acquired it or contributed to it, such as by being an authorized user on your credit card. If you don't pay the debt, the creditor can sue you and even try to collect on your share of jointly owned assets.
Some times divorcing your spouse is easier than divorcing your credit card debt. Even if your divorce decree orders your spouse to pay the credit card debt in your name, you are legally responsible for paying the credit card debt.
In Texas, which is a common law state, courts will likely hold you responsible for credit card debt in your name and jointly liable for credit card debt in both names. Even when you are not contractually liable for your former spouse's credit card debt, a judge may order you to pay a portion of it, particularly if the debt was incurred for items necessary for the household. But if you don't pay, the only remedy your ex spouse would have is to sue you in family court to enforce the order to pay. For that reason, judges in Tarrant County often order the parties to pay the credit cards in their respective names and make up for that award of debt by the award of other community property. A judge may distribute responsibility for the debts in a way that is different from your contractual obligations to pay, but the judge's order does not change your contracts with your creditors.
If a credit card is in your name, the creditor can come after you if your spouse does not pay a debt as ordered. The same is true for your spouse's debts that you are ordered to pay. Your recourse is to pursue your ex in court. This can be a difficult and expensive journey full of headaches.