In Family Court, you are only entitled to a "free" court appointed attorney if you are looking at jail time.
If you are not paying your child support and an attorney files "contempt" on you, many will "waive" jail time so that you cannot get a "free" court appointed attorney. The family attorney does not want you to have an attorney representing you since it's much easier to hold you in contempt since you probably won't know the law.
If you want a pro bono (free) attorney there are limited non-profit organizations in the State of Texas to help you. Look on the State Bar of Texas website or a list of pro bono organizations to help you.
You will have to meet their qualifications to be represented by one of their attorneys.
Each non-profit has different guidelines and they often change due to their funding sources. Some years they might receive a grant to handle specific cases (veterans are popular right now) and when the funding ends then that program goes away.
Most require you to send in your income and be able to provide identification. Some require you to come into their office to fill out the application. Some non-profits do not handle certain types of cases (such as appeals or contempt). So call around for their current policies. Visit their websites for more information.