Politicians have held office while serving time in jail for crimes. Committing a crime before being elected, is thus obviously no barrier to holding office in all cases.
As for her calling herself a single mom, it is not only not a crime, it is not even objectively untrue as "single mom" could be taken to mean "raising one or more children without significant financial or time contribution from a partner" instead of "unwed mother".
This reminds me of the fuss about the character Murphy Brown being a unwed mother -- IMO no one that hires a full time nanny can accurately be called a "single parent". They may not be married, but they obviously have assistance in providing the attention that a child needs, and have no need of any financial support.
Conversely, a married parent where the partner is unavailable (in jail or deployed in the military for instance) could fairly be called a single parent: they are raising the child without the aid of another parent.
Also, I would not call either partner a "single parent" if they were living together and raising the child together but just weren't married. Even if they were not romatically involved, but were just jointly raising the child (i.e. providing food, shelter and attention).
In short legal martial status is not a defining characteristic of "single mother", which means that even if she said it under oath, it would not neccesarily be perjury.
Note: on reading your link, it seems that the "lie under oath" would be about when she divorced her first husband. That might possibly be sufficient to get a conviction for perjury, but I wouldn't call it a slam dunk case. You'd have to prove beyond a reasonable doubt that she didn't simply mis-speak or mis-remember. Absent a conviction it certainly isn't sufficient to bar her from office. I don't know if it would be legally sufficient under Texas law if she was convicted.
Edit: Did some reading, it looks like Texas law prevents felons from running for office, simple perjury is a misdemeanor.