As you probably already know, public intoxication is a class C misdemeanor in Texas, punishable only with a civil fine. It's the criminal equivalent of a traffic infraction. So the worst you're looking at is a fine. There would be a record of your conviction if you pled guilty to this, that record doesn't go any higher than the local court level. By that I mean,there's a copy of this filed with the court, but it should not show up on an official state or federal background check.
According to Sec. 12.03(c) of the Texas penal code, a conviction on a class C misdemeanor doesn't impose any kind of criminal or legal liability on the offender. So it is not in the state or Federal criminal data base and you wouldn't have to disclose it.
On the other hand, a copy remains in the courthouse and it is not under seal. That means it's public record and could potentially be found by private background check companies. It won't turn up on your RAP sheet, but someone doing a thorough background check could potentially find it anyway.
Texas will not remove convictions from your record unless they have first been pardoned by the governor, even for a Class C misdemeanor. So you may want to retain a lawyer and see if he could negotiate a diversion type of disposition. This is where you have a period of supervision, pay some fines, do some community service and stay out of further trouble with the law.
Complete it all successfully and your case gets dismissed. Diversion is more work than paying a fine, but in the long run it's worth it because If you can get diversion, then your record can be expunged. If you can't get diversion but can get a deferral and community supervision, you could get an order of non-disclosure, which means the record would be sealed.
If you want to fight the case rather than take a disposition, since he didn't give you a breath the and the only evidence that you were intoxicated is his testimony, then your worst case scenario is still only a fine. Still, to fight it, you should retain a lawyer for best results.