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You are on the right track with the letter.
Someone spreading false statements about you which are damaging to your reputation and could result in the loss of a job is technically defamation. You may sue the people involved if you have evidence that they made the statements and that the statements are damaging to your reputation. The statements must be false, or they can win a lawsuit.
That being said, in your letter you need to specifically state that they should CEASE AND DESIST all further untrue statements about you or you will immediately take legal action.
Not disclosing this matter to HR is probably not a good idea. The actions of co-woerkers could be seen as creating a hostile work environment and they may be disciplined by the company for doing that.
Are these co-workers making the false statements? If so, why not tell HR or a supervisor?
The reason I did not go to HR is because I don't want anyone to loose their job either. But they are creating a hostile environment. They are telling people and these people are taking sides so therefore I have been getting dirty looks, ignored completely, talked about, etc. I don't want to go to work feeling uncomfortable, I've been there for 15 years and never had any problems before. I am not a trouble maker or like drama. I just didn't want to make the situation worse.
It's HR's job to stop things like this from happening in the work place. I think you would be wise to go to HR and state your complaint and tell them that you are going to sue these people for defamation and that it is creating a hostile work environment. Being active in protection of your rights is always the best thing to do.