I was told that DAs and other prosecutors are typically prohibited
from engaging in volunteer, pro bono work
You have not really understood the linked answer and are oversimplifying its analysis. I'll restate the analysis in a different way in this answer in the hope of providing greater clarity.
The key to understanding that question is that the question presented there was: "Are DAs and other prosecutors prohibited from being defense attorneys, such as in a pro bono fashion?" Having a prosecutor do criminal defense work presents conflicts that other types of pro bono work frequently would not.
Does this mean that public attorneys can engage in pro bono work in
Texas? If so, are there strings attached? For example, would a public
attorney be able to represent their friends and family so long as they
don't charge for the work?
These questions don't have a single global answer and must be analyzed on a case by case basis. It depends upon whether the pro bono work constitutes a conflict of interest with the public attorneys' regular work (i.e. what kind of legal work is done pro bono and for whom) and it depends upon the policies of the lawyer's particular employer.
There are two distinct questions presented, neither of which are materially different in Texas than they were in the previous question.
One is when doing so would violate the professional ethical obligations of a lawyer, which is what the Texas Bar Association statement is talking about.
The other is whether it is permissible in a manner consistent with the contractual employer-employee relationship of any particular Texas attorney working on a full time salaried basis for a particular employer. This is a subject upon which the Texas Bar Association statement is silent.
Professional Ethics Considerations
As noted above, one question is whether pro bono work by a prosecutor in the specific subject matter of criminal defense would constitute a conflict of interest. The answer to that question is "usually yes" for the reasons explained in the linked question.
One can also easily imagine other circumstances where pro bono work could pit a salaried government lawyer against his employer, such as a state attorney general's office lawyer doing pro bono civil rights litigation against the state government.
But, not all pro bono legal work constitutes a conflict of interest.
For example, a prosecutor doing pro bono work helping low income people prepare documents naming legal guardians for their children in the event that they predecease their children (the very first pro bono case I took on as a Colorado attorney) would be very unlikely to be conflict of interest for the prosecutor, even though a pro bono criminal defense case in the county where the prosecutor practices would be an obvious and direct unethical conflict of interest.
Similarly, judges frequently do pro bono work, where authorized to do so by the court systems in which they work, in the areas of legal education for non-lawyers (like High School Mock Trial competitions or lectures to school students) and by providing mediation services to indigent parties outside the judicial district where they practice, that are specifically calculated to avoid conflicts of interests, or appearance of bias or impropriety.
Employment Relationship Considerations
The second issue is whether "public attorneys can engage in pro bono work in Texas?" in a case with a subject matter that does not otherwise present a conflict of interest.
Public sector attorneys are not prohibited from doing so as a matter of professional ethics. But, that isn't the only consideration. The other consideration is whether doing that pro bono work violates the employment policies of a public sector employer of a lawyer in the usual case when a public sector lawyer is employed on a salaried basis for the full efforts or something similar, of the lawyer. This is basically an issue of contract law, not of professional ethics.
The way this is usually dealt with in private law practice when a salaried attorney works for a law firm is that the private salaried lawyer needs to obtain law firm permission to do pro bono work and that pro bono work is often structured so that the pro bono client's lawyer is the law firm itself and the lawyer in that firm doing the actual pro bono work is often just one agent of the firm as a whole that actual represents the pro bono client. Frequently, salaried lawyers in private law firms have quotas of billable hours which they have to meet as a condition of employment each year at the law firm, and frequently a private law firm's pro bono program allows the salaried lawyer to count hours working for the law firm's pro bono clients towards that annual billable hour quota.
Incidentally, there is nothing magic in the absence of a conflict of interest, about limiting "moonlighting work" to unpaid pro bono work.
For example, many U.S. law schools expressly authorize salaried full time law professors who are admitted to the practice of law to have a paid private practice up to a certain number of hours per year as a fringe benefit in addition to the salary, akin to the common practice of allowing professors to retain for themselves textbook royalties for textbooks written while the professors were being paid a salary to do scholarly work including writing textbooks.
The pro bono program structure used in private law firms for their salaried employers isn't as workable for a salaried attorney working full time for a governmental employer. This is because allowing a government salaried employee to work for free for a private citizen on something that doesn't involve public business of the government would functionally be an indirect appropriation of legal salary funds from the governmental entity to the pro bono client without receiving authorization from the elected representatives of the government in question.
So, in the public sector, a more common arrangement would be to provide that every salaried public sector lawyer working for that government is permitted to do up to X billable hours of pro bono work per year for their own, personal moonlighting private practice clients as a fringe benefit of employment, much like paid time off, if it is permitted at all. And, other public sector employers, which are not so generous, simply prohibit their salaried attorney employees from doing pro bono work at all in order to avoid having to deal with and think about the issue.
This doesn't arise very often in the case of lawyers hired by small local governments or on a contract basis by a larger government for a particular specialized task, as they have their own practices and have not agreed to provide substantially all of their professional legal work to their employer the way that a salaried lawyer would. So they are more akin to private sector law firm lawyers who simply have an additional set of conflict of interest considerations as a result of having a particular local government as an employer.