it is a great thing that you are recovering quickly. your health is more important than the legalities of the situation. sounds as though you have the right attitude and are proactive in getting better - that helps. i often tell injury clients that they are better off if their case is a small one because it means there isn't anything seriously wrong. what the insurance companies decide amongst themselves re fault is not binding on you. they often divvy up the fault as a matter of compromise. your lawyer will put together your records and other medical documentation, submit it and hopefully the opposing insurance co will be sensible and reasonable. good luck
It is great that you are feeling better. However, the most prudent thing to do is to get the most comprehensive evaluation, especially since it covered by under the auto policy. Often the injuries are dormant and not addressed immediately after the accident. Once they manifest themselves, you do not want to be in a position where the benefits and legal recourse is no longer available. Once you get the "clean bill of health" you may consider not pursuing the claim against the negligent driver.
The above positing is not intended to constitute advice and does not create attorney-client relationship.
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