I suggest you speak to an attorney first, and wait till you go to court. In most cases the attorney can provide your true name to the court without any major problems. A Third DUI carries a mandatory minimum sentence of 120 days.
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You should contact AND retain DUI counsel immediately. Your sentence will depend on a number of factors, now of which are mentioned in your question except for the fact it is a 3rd DUI.
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Wait for court, but I agree you should contact and retain counsel well in advance.
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If you can afford an attorney, you should hire one. If you can't, you should go to court and seek the services of the public defender. This type of matter should not be discussed further in a public forum. Giving a false name that could result in the prosecution of another person can be charged as a felony. Not sure what kind of mistake was made. The jail has your fingerprints, so if you have any identification or prior jail contact with fingerprints, your identity can be established. The jail that you are looking at for a third DUI depends on a number of circumstances, incluidng wether you are on probation for any of the priors. Many attorneys, judges, and district attorneys are not aware of provisions of the vehcile code that allow for a 30 day minimum jail sentence with a 30 month alcohol program. I have obtained this for clients. It usually involves a process of educating the courts and district attorneys, together with getting the client into treatment in advance of their court date, so that there is justification for leniency.