This sounds like a mostly 50-50 fault situation to me, though the backing driver might be slightly more at fault. But the other driver's insurance is pursuing you because you are uninsured and thus there is no one to defend you - they are hoping you will just pay. If they take you to court, you will have to pay for a lawyer to defend yourself. Which sucks, because you really shouldn't pay them a penny. I would tell them to pound sand, you were not more than 50% at fault which means they can't recover from you under TN law. Then wait and see if you are served with court papers.
Now, you may still lose your license for driving uninsured, but that's a separate issue. And if they decide to sue, you will need to defend yourself. But personally, I'd take that risk since they really don't have a case.