The cases always start at the Magistrates court, even murder! The question is, will it be dealt with there.... you can elect trial at Crown Court or the CPS, if they feel that the case is serious enough, can ask for it to be tried at the Crown Court.
i would suspect that CPS would be happy for a case like yours to be dealt with at the Magistrates court.
It depends as to how far you drove, in what lane and at what speed on a flat tyre. There could certainly be a case made out for dangerous driving if your tyre was shredded and you drove any more than a minimal distance to get to the hard shoulder or an exit.
As I said before, your best bet is to get a solicitor to try to have the dangerous driving dropped and the court to be told about the tyre as an aggravating circumstance of the drink driving.
You could ask the duty solicitor at court on the morning of the case, you would qualify to speak to him / her and have representation in court free as dangerous driving carries a risk of custody, but that would not give much time for the consultation process to take place,