Good evening and welcome. I am very sorry to hear that you were let go and agree that it sounds quite unfair.
Regarding your question, "wrongful termination" does not mean what most non-attorneys think it means. It does not refer to a termination which is "wrong" in the sense that it was unfair or should not have happened. Rather, it refers only to terminations which are motivated by legally protected traits (e.g. race, religion, gender, disability) or a legally protected activity (i.e. filing a workers compensation claim, taking FMLA leave). Only then does a termination because legally actionable as a "wrongful termination." Termination for any other reason is legal in Texas, regardless of whether the basis is fair, reasonable or even TRUE. This is because Texas follows the doctrine of at will employment, the idea being that either the employer or employee can terminate the employment relationship "at will" without having to show cause.
Regrettably, this means that the facts you have described would not give rise to any legal causes of action against your employer. The EEOC would not have jurisdiction, either, as the EEOC only deals with terminations motivated by legally protected traits, as defined above.
You can apply for unemployment benefits as you search for a new job, but unfortunately your only option here is to appeal your termination with HR (note, this is not a legal "appeal," just an attempt to get your employer to reconsider) or to start looking for new work.
I hope that you find this information helpful and am genuinely sorry if it is not what you were hoping to hear. Please do not hesitate to let me know if you have any questions or concerns regarding the above and I will be more than happy to assist you further.
If you do not require any further assistance, please be so kind as to provide a positive rating of my service so that I may receive credit for assisting you. Very best wishes moving forward.