Your 6th Amendment right to an attorney does not attach until your arraingment. However, they did allow you to contact your attorney.
Many people who are not lawyers get Miranda all wrong. They think that if the rights aren't read that their case is lost. In fact, Miranda, important though it is, has a far more limited application. Miranda stands only for the fact that after an offender has been arrested he can refuse to be interrogated about the arrest he was arrested for. If the police do not give warnings before they interrogate then any confession or statement made by the defendant cannot be used against him at trial
. In some cases, where there is no interrogation
, for example, Miranda warnings don't even apply.
I don't see anything wrong with the warnings as your name is not written into Miranda nor is your name anything they are required to read to you. However, I am getting only the tip of the iceberg and don't have the benefit of the information I would see and hear if I were actually representing you on this case. So when you have your lawyer, and he discusses the strengths and weaknesses of your case, if you decide you want to fight these charges, he will be able to move for a suppression hearing to address all of the things that you are concerned about above. He should be able to give you a good idea of whether there are strong suppression issues or not and whether you can get the blood test and any statements you made thrown out after a hearing.
Again, I don't see anything and wouldn't count on that, if I were you, but like I've said, I only have a small piece of the information. As you have a lawyer after he sees the court
papers and confers with the prosecutor and sees some discovery, he'll be able to give you a more detailed assessment.