It is not too late. In CA, most personal injury actions like yours have a two year statute of limitations. In cases like this, I work out an agreement with the client to make sure they get at least what the insurance company offered before they hire me. Usually it is more.
It's hard to answer if you can have the same outcome with a public defender. However, you should try your best to be cooperative as possible with your assigned public defender so that it will be easy for the both of you to work together and get the outcome that you are expecting. Good luck
The booking did not took place right away because you were taken to the hospital. I suggest that you hire a lawyer before you appear in court on the date indicated on the letter you received. Good luck
If you have the financial capacity to hire a private lawyer you must do it. Public defenders are good lawyers too, however, they do not have much time to focus on your case because they have a lot of cases in line. Remember, that your career is at stake here so you must do every effort to clean you record. Good luck