That's not quite right. They may send you a form asking you what happened, but you are under no obligation to tell them and by doing so you'll only incriminate yourself. If they want to question you, it needs to be under caution at a station and with the option of legal counsel. Doing anything else is utter madness and the quickest way to get major points.
What's the worst that can happen if you don't have a solicitor? You cough up to everything, that gets twisted into whatever they like and you end up with 6pts + a massive fine for leaving the scene and DWODCAA. What's the worst that can happen if you have a solicitor? They tell you not to say anything at all and it's up to the police to prove what you did wrong.
I have complete respect and faith in our justice system, I really do, but no way in hell would I cough up info freely in something like this. You seem genuinely remorseful and regret what you did, and you've been honest enough on here, so I doubt you're a complete idiot else we'd have all given you hell for it Lesson learnt, why make life harder for yourself?