Hi SurreyGuy, I have read your question and all the answers with interest. I too was in a very similar situation, dismissed unfairly from my job, a newspaper company back in 1993. The company, paid me to keep me from taking it to tribunal. The company told all the workforce that I had been dismissed and made me out to be a bad apple.
I was very stressed out by the whole incident and it took me a long time to regain my confidence. But I stayed in my next job for 10 years, was very happy and progressed well in the company. I now run my own business and ironically use the company who sacked me, to place ads for my company, how sweet it is to hear them suck up to me to gain my business!!!!
However, I completely understand your anger and frustration and your need to 'clear your name' with your old work colleagues.
I would suggest that you wait until you are relatively well established in your new job and your reputation is strong in the new company, in this way they will not pay much attention to any further attempts at contact from your old company.
My advice would be to send these letters, disguised as a chatty update from you, telling them how wonderful you are doing in your new job, how well things have worked out for you and how glad you are that you left your old company when you did, or you would never have got the new job at such an opportune moment!
This will really pee off these people who will realise that all they have managed to do is to help you gain a far better job in a much nicer working environment. You could finish off by 'thanking' them very sincerely for their assistance in helping you achieve this!!
If you are clever enough in writing these letters, the company cannot accuse you of breaking any agreements as you have only sent a friendly letter, also you will have the satisfaction of knowing that they are fully aware of how happy you are now.
Hope this is of some use, all the best, Sue