No. That is his obligation, not yours.
This answer is intended for informational and educational purposes only and should not be considered legal advice nor forming the attorney client relationship. This attorney is licensed in Texas.
No, you are absolutely in the clear on this. The Texas Family Code explicitly states that the income of a spouse of a person who owes child support for the child of another relationship cannot be used even to calculate the child support is owed. Only the actual parents of the child can be ordered to pay child support.
Just as a "by the way", there is no "legal separation" in Texas--you are considered to be married up until the day you are divorced. If a divorce petition has been filed, you're not living together, and temporary orders have been entered in the divorce case, you might as well be legally separated for most purposes, but not all, so just keep that in mind should it come up.
Another "by the way" issue you might be facing is that if you've built up a close relationship with your husband's child, you don't have any legal right to continue to see the child, which can sometimes be very hard on both the child and the stepparent. If you'd like to maintain contact, trying to maintain a friendly relationship with both your husband and the mother may be the only way to achieve that. Good luck.
This answer is intended only to give you general information on the subject you've asked about and does not constitute legal advice or establish an attorney-client relationship. If you feel you may need legal representation, please contact me by phone at (409) 750-0757, or by email at firstname.lastname@example.org, and I'll be happy to provide you with a free consultation about your case (up to 30 minutes) and an explanation of my fee schedule, payment options, and the different levels of representation I offer to fit your specific legal needs and budget.
Sign up to receive a 10-part series of useful information and legal advice about the divorce process.