You were NOT arrested for a class C misdemeanor where someone talked to the DA. If you were arrested for a Class C, then they talked to an assistant city attorney or an assistant district attorney (in a JP court). The charge IS a ticket level offense so there is no "dropped to a ticket" for a class C.
If it was dropped from a Class B to a Class C (ticket) and you got deferred, then you can expunge it. If it was dropped from a Class B to a Class C & you paid a fine, then it is a final conviction and will stay on your record.
Call your attorney who represented you and find out what happened, and / or get a copy of your plea papers from the court at which you appeared (which is likely a county court - not a JP or municipal court).
Make an appointment with that attorney and sit down with her and the case file. That is your first move. I wouldn't go in there accusing her of lying. You want her to help you straighten up whatever misunderstanding or mistake in the court record. If the situation doesn't get resolved that way and you are convinced she did something unethical, seek the opinion of another criminal defense attorney. Good luck to you.
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Depending on the specific facts of your case, you may be able to get an expungement, but I do not have enough details to be able to say if you can or not. www.giardinolawfirm.com.
There are several legal conflicts in your question that make it clear you do not know what you were charged with or the disposition of that case. You can go to the court where your case was assigned and request a copy of the records from the clerk. If you do not understand those records, you need to take them to a local defense attorney who can explain them to you. You can also set up a meeting with you original attorney and ask her to explain everything to you.
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Nobody here can tell you whether you're going to be able to get your job back. Texas is an at-will employment state, which means that an employer can fire you for any reason (or no reason). They don't have to have some legal justification for doing so. So if they want to keep you, they will, and if they don't, they won't. As to this "certified letter", I have no idea what you're talking about on that. Sending it certified mail doesn't make any sense in that context. Possibly they said they want a sworn affidavit from her? In which case, I rather doubt she'll do that. If you're willing to sign something waiving attorney-client privilege, she may be willing to talk to them, and maybe even write something briefly explaining why there might have been a misunderstanding on your part, but she obviously cannot swear to what you believed (you could, but she can't). All she can do is say what she told you, and there's no reason for that to be sworn. If it is a matter of an expungeable case, that's a matter of frequent misunderstandings. There's a very big difference between "You can someday get this off your record" versus "This is now off your record". Expunctions are separate legal proceedings, aren't cheap, typically can't be done without some time delay, and are often basically forgotten about. If that's what happened, it's not your attorney's fault, and I doubt she's going to be willing to take the blame for it.
I agree with the technical information provided by the other attorneys. I also want to add a practical approach: When you contact the attorney who represented you in the prior criminal matter, ask for her assistance in communicating these details to you. It isn't enough to just obtain copies of the court files on your own if you cannot sufficiently explain those files to your employer. An attorney with criminal experience needs to explain this to your employer. Hopefully that explanation will preserve your job for you.
It is true that Texas is an "at will" employment state which means you can be fired for no reason at all (unless you have a written contract, or other terms of employment that protect you). So you can't find your protection there. The real issue here is whether the crime itself is one which your employer is willing to accept in terms of keeping you on staff. If they are not crazy about the crime, they don't have to keep you on payroll.
Get more information; elicit the help of your prior lawyer. Make it clear (IN WRITING) to your prior lawyer that your employability is on the time.
Best of luck