No, the State can't come after you for the child support of your husband's child from another woman.
This is not to be considered legal advice nor does an attorney-client relationship exist.
No, you are not responsible to pay child support for someone else's child. However, keep in mind that the state will impute income for your husband, so if he is not paying child support he will accrue child support arrears.
You are not responsible for the actual support but if H cannot support himself and pay support the court can consider your income and finances as available to him in imputing income to him. Keep your accounts separate if nothing else!!
This answer was provided as a courtesy to you and no attempt was made to establish any type of attorney/client relationship.