Is there such a thing as free divorce for very low income single moms?

The best answer is "kind-of" and "it depends."  Texas does have a procedure where anyone who is indigent can file an affidavit of indigency at the same time they file for divorce.  Usually the clerk will walk you to a judge and if the judge finds that you cannot afford to pay the court costs associated with filing for divorce, then he can authorize the waiver of the filing fee.  This can usually save you around $300, plus or minus your jurisdiction.  More and more forms are available online that you can use to draft the original petition for divorce, the affidavit, and the final decree.  These are the three forms that you need to complete a divorce.  If you are comfortable filling out your own forms, then it is possible to do the entire divorce yourself for free, as long as the judge approves the waiver for filing fees.

You can also do divorces very inexpensively with some cost saving options.  The first is to file your own petition and affidavit, but hire an attorney to draft the final decree for you. When you file your petition for divorce-- if you leave something out or you mess it up, you can usually just file an amended petition with no-harm/no foul.  The final decree is different. This final order controls who gets what and when your ex- is required to pay child support.  If this document is done incorrectly, you could loose valuable rights.   Some attorneys will do this for a small fee, which is worth the investment if you have any custody or property issues.  A third option is to draft the forms yourself, and then try to have an attorney proof read or review your work at a local legal clinic.  Some counties offer free legal clinics once per month at a local church or libary.  A third option is to go "halves" with your spouse.  An attorney can only represent one of you, but if everything is agreed to upfront, then some attorneys will help work the two of you through the process to finalize the details-- but at half the cost to each of you. 

If you are very low income, you may qualify for legal aid.  Every county has its own name for this type of organization:  "Legal Aid", "Legal Aid Society," or "Legal Services."  Often they are run or funded through the local bar, a local United Way program, or some other non-profit organizaiton.  So call your local United Way to see if they have an affiliate that does divorces.  Also call the local district clerk to see if they have any information on the programs in your area.  I've also seen postings at churchs and local libraries.  And... of course... do a search on the internet.  If you qualify, then often the legal aid organization will do the entire divorce for free or a nominal charge.