Thank you for the information and your question. First, if you believe the employer is making false statements of fact about your employment history, you have the right to hire an attorney and file a defamation of character action against them. If you go that route, your attorney can find out more information than a private investigator can because they will depose (under oath) the employer's personnel that either are involved in providing the references or that might know about any statements being made. That though is your choice.
As for interviews, that is more difficult. The last thing that prospective employers want to hear are complaints or troubles with prior employers. Right or wrong, that sends a signal that the employee might be contentious and that is the last thing the employer wants. So, it is important to try to be diplomatic when explaining why you left a prior employer. I can't tell you what to say because you have the facts, I don't have all of them including those that the employer has, and you will have a sense for how much a particular employer may need to know. Usually, if there is an opportunity to interview and you have written that you resigned your prior job, you can note that you will be glad to discuss the circumstances of your leaving. At that point, if you do, keep in mind what I mentioned about complaining about the employer and balance your sharing of the reasons with what the employer really needs to know about your work and employment. Also, keep in mind that defamation works both ways, so you want to stick to facts that can be proved.
Please feel free to ask for clarification if needed. I will be away from my computer for much of the day but will respond to your reply as soon as I am available. If no further assistance is needed, then if you could take a moment to leave a positive rating in the ratings box above, I will receive credit for assisting you today. Thank you