As is usually the case, your first step should be to contact a lawyer in your jurisdiction. Advice in these matters can be extremely fact specific. As has been often stated here - most lawyers do not charge for an initial consultation. If you can't find a local attorney any other way, you can contact me directly - I primarily practice in Dallas, but I practice in Plano occasionally.
Your second step is to carefully read the documents you received when you got out of jail. You cannot simply "pay the fines" to get out of jail without first being convicted of the charges. I know this is Texas, but even in Texas we don't sentence people before convicting them. This may not have been explained to you.
Unless you appeared before a judge, waived a jury trial, and pleaded guilty, you couldn't have simply paid the fines to get out - instead, you posted cash bonds. There is a big difference. If you really did plead guilty and pay the fines (which I think is highly unlikely) the case is over and you have no recourse. If you posted cash bonds, you can still request court dates and challenge the charges. You must request a trial within the given time period (again - read your documents) or contesting the charges becomes much more difficult. Lawyers who handle these cases know this stuff - so they have a huge advantage over non-lawyers and lawyers who never handle PI cases.
To prove you were guilty of PI in Texas, the police and prosecutors must prove that you were "intoxicated to the extent that you were danger to your self or others." Sadly, Public Intoxication arrests in Texas are sometimes made simply because someone "argues with the police." Some police will make an arrest in that situation and justify it by stating "Anyone who argues with me is a danger to themselves." Fortunately, many times an officer making such a bogus arrest will not appear in court or testify against you.
You may be able to successfully challenge these cases in court, but without a lawyer, I don't like your chances. If you can convince the prosecution to give you "deferred adjudication" on both cases, upon successful completion, you may petition the court to expunge (erase) both charges. But you need to be very cautious - if you screw it up, the arrests may be on your record forever. For these reasons, most people would hire a lawyer. You have your entire future at risk.
If you want the job done right, you should bite the bullet and hire a lawyer. This stuff may not be rocket science, but it's well beyond the average non-lawyer.
Best of luck to you!
I agree with John M. Gioffredi's thorough answer. My guess is, based on your explanation, that you did not post a bond but instead simply paid the tickets. (Next time, post a cash bond and get your money back after the case is over. This will give you an opportunity to contest the charges as it sounds like the officers in your area are worse than most.)
If you are still within the time limits, you may be able to ask for a trial in misdemeanor court (one step above justice of the peace or municipal court) if the court in which you were convicted is not a court of record. (Meaning that they do not have court reporters and you didn't waive one but instead contested the charge and lost.) OR, if you are still within the time limits, you may be able to appeal to the higher misdemeanor court.
In the future, if you cannot get a dismissal and you do not want to chance a trial, ask for a deferred adjudication and then you will have the chance to later seal your record once you serve out the time for the deferred.