Sounds like there are inconsistencies in the police report and lab work that can be used to your advantage. Also sounds like they are claiming that the blood was a forced draw.....something your attorney can look into further. Good luck on your case.
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I don't think you should be seeking advice on what to do here. If you have a public defender, trust that they will look into and prepare an appropriate defense. Your Public Defender is a licensed attorney. If you don't trust they will do their job, than start looking for a private attorney. Without a solid understanding of all the facts, anything we could offer would be speculation.
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You are going to be requesting the services of the public defender and this is an issue that you need to address with them. The issue you will run into is the PD does not represent you in the DMV action and the DMV based on the police report will suspend you license for a year, unless you hire a competent attorney to fight for you.
Without knowing the test results, it is hard to suggest a defense. First, the breath test seems unreliable on its face as the cop says at least once, you did not take it. Were you read a very long admonishment before the blood draw that explained if you did not take a blood or breath test that you would lose your license. It is not a refusal if you cannot complete the breath test and have to go to a blood test unless you say you aren't doing either. Also, the breath machine should have the failed or completed test in its memory and your lawyer should be asking for that in discovery.
Any comments offered are not intended as legal advice. This attorney does not know the specifics of the particular case sufficiently to offer legal advice. You should hire a lawyer who can evaluate the details of your specific situation before offering advice.