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Thanks for your question.
I would very much recommend that you find out about EMDR (Eye Movement Desensitisation and Re-Processing), a therapy that can guide your subconscious mind back to the memories of long ago in the car and release the memories, emotions and thoughts that are 'stuck' there.... I will explain.
When we are asleep (as you probably know), we process anxiety from the day away during REM (Rapid Eye Movement) sleep, where information passes to and from the left and right sides of the brain rapidly in order to process the stuff away. If we have experienced a highly emotional situation, (skidding in a car on ice would be sufficient) then irrational beliefs can become stuck in the subconscious mind in a place where they aren't processed at night by REM.
The role of the subconscious mind is always to protect the self, so the subconscious mind begins to create negative feelings at a time when it feels danger may be around the next corner - for you, driving the car. It can become so negative that the person avoids the activity - just as you have avoided driving. The subconscious 'gets its own way' because it has a powerful effect on us. So your world is changed by these beliefs, but they still remain in the subconscious mind, triggered by normal stuff in everyday life - then you get the symptoms that seem to come from 'no-where' - tearful, low mood, stress.
An EMDR therapist will enable your subconscious mind to let this all go so the future is more rational and manageable. It works by simulating REM sleep by using a bilateral stimulation (tapping, lights, electrical pulse, etc) whilst thinking of the old stuff. Because it all happens in the subconscious mind, you don't need to be able to rationalise your fears at all - they may even be linked to something different that you're not consciously aware of.
There are no guarantees in any therapy (as you will know) but this one can be very powerful and relatively quicker than talking therapies, but the subconscious mind does the hard work for you. This is a bot of an odd therapy, when I learned it, it seemed most bizarre. But having experienced it, and used it with clients, I have seen just how amazing it really can be.
I strongly suggest that you search for a local therapist on the internet. Try http://www.emdrassociation.co.uk/ You might be able to use the experience in your training or even decide to specialise in trauma work. I wish you well with your issue and your studies.