I agree with my colleagues. You have done a great job outlining your question and the underlying facts. I am not familiar with Texas small claims court. In California, the small claims court has significant discretion to provide for an equitable resolution. If you erroneously executed the financial responsibility documents, the court will hopefully agree with your position. If the debt (including the costs that will be added) is not significant, you should roll the dice to see how it plays out in small claims court. If the debt is significant, it may be prudent to spend an hour or so with a local attorney going though the facts so that you have a more detailed and thorough understanding of the facts and the application of law.
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Given your additional information, I am inclined to agree with you. My concern, however, is that it appears that you signed to be the "responsible party." That generally makes you liable for the debt. In order to protect yourself, you could have signed your name, as agent under the POA. Hindsight is 20/20, however. If they are suing you, you need to respond. I am not sure if you can use attorneys in small claims actions in Texas. In Michigan, you cannot.
You might benefit from purchasing an hour or two of attorney time, regardless. If the debt is significant, the attorney can help you to present your position in the best light possible.
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You did a good job of presenting the facts. I share Mr. Frederick's concern that you may have signed as a responsible party. It is impossible for an attorney to express an opinion about the legal effect of a document without having reviewed the entire document. Small Claims court in Texas is designed for non-lawyers. Lawyers can appear there but in my experience they often do no better than non-lawyers because the rules of procedure and evidence don't apply. They are not courts "of record". That means that the loser in Small Claims court can appeal to the county court at law (short time for appeal) and the whole case is tried over again in the county court at law where a lawyer can be of great help. Another alternative is for you to use a lawyer to "coach" you prior to your trial in Small Claims court. An hour of an experienced attorney's time would do you great good. But, you must answer the lawsuit on time or the court will sign a default judgment against you.
DISCLAIMER: This is not specific legal advice and does not establish an attorney/client relationship.