In order to prove that you committed the offense of public intoxication, the State of Texas has to prove beyond a reasonable doubt that you were in a public place and intoxicated to the point where you posed a danger to yourself or others. The police do not have to perform field sobriety tests or give you the opportunity to take a breath or blood test (though it certainly would help you if you requested some of these tests and the officer(s) did not perform them). It is an opinion crime and the State can be wildly speculative about the "danger to yourself or others" element of the crime.
Whether or not you have a good chance of winning really depends on information you haven't included in your post. You would have to convince the jury that either you weren't intoxicated or you weren't a danger. There are no additional fees for representing yourself, since that is a right you have under the 6th Amendment. There will be additional fees if you are found guilty because you requested a trial.
A lawyer would be able to help you get all the testimony in that helps prove your version of the events because he or she should be proficient in the Rules of Evidence and cross/direct examination. The prosecutor will be a trained lawyer and will not go easy on you simply because you are representing yourself.
Disclaimer: This answer is provided as a public service and as a general response to a general question, it is not meant, and should not be relied upon as specific legal advice, nor does it create an attorney-client relationship.
Mr. Walcutt's answer is thorough & correct. In addition, you mentioned that the officers did not read you your Miranda rights. They only way that would come into play is if they attempted to offer any statements that you made at the time of your arrest. While they are supposed to read you your rights, the remedy is to suppress the statements if they fail to do so.
Your chance of winning depends on the strength of the testimony. While the burden is on the State, you already know that the officers are going to testify that they considered you to be a danger to yourself. If you testify that you were willing to leave - and that meant in a car - the officers would certainly stand a greater chance of succeeding in your prosecution (and the State would play up that you would be dangerous to the public behind the wheel of a vehicle.)
A public intoxication is an alcohol related offense. You would want to do your best to keep this off your record, as law enforcement would be able to see the offense in the future if they stop you for any traffic offenses, and it would just appear as a strike against you.
If you go to trial and win, then you will not be required to pay anything. If you go to trial and lose, you will have to pay court costs and some sort of fine (under $500). If you plead guilty or no contest, you will be required to pay court costs.
It's not a great idea to represent yourself in traffic court. You do, however, have the constitutional right to do so. I would recommend spending an entire day in the traffic court to observe how judges handle self-represented defendants to get an idea of how irritated they get at amateur lawyering.
If you really want to represent yourself, you should purchase an hour of a traffic attorney's time to have her or him coach you on how to handle things.
A better choice is simply to hire an attorney.