How do I prove I am being slandered?

I used to work for a hospital in New York City. I was bought out of my position 7 months ago because of an email I sent out. It was not slanderous, illegal or inappropriate in any way (i checked with a lawyer), it just ticked off my boss that I sent it. Because I had a clean record, no negative reviews, no write ups, nothing bad in any way in my file, I negotiated a termination contract with the hospital that clearly says that it would go in my perm record that I resigned and nothing bad would be said about me to discourage future employment elsewhere. During negotiations, HR told me that I had been repeatedly warned that I was in danger of losing my job, which is entirely untrue, but since it wasn't in writing, I didn't think anything of it. Second, when I went to apply for unemployment benefits, I was denied because the hospital reported that i was fired for insubordination, again, untrue, slanderous and a breach of the contract as there was a clause in there stating that they would not do this. HR ured me they cleared it up. Well now 7 months later, I am convinced that they are slandering me and am unsure what is in my HR file. I have had numerous successful interviews, and even a verbal offer, then suddenly, after a reference check, the companies disappear and won't give me the time of day. I was even flown out to California to meet with a company, which went very well. However, soon afterwards, I was given a very polite f-you. This was enough time afterwards to check my references. I have also been contacted by three other hospitals in the area for positions, only to have them refuse to talk to me when I return their call, which was normally enough time for them to call my old hospital. What do I need to do to prove slander, defamation and breach of contract? I have already caught them lying twice, I have no doubt they are doing something unethical again.