You have a contract with your employer's medical insurance, whereby they will cover your expenses from this accident; following the most likely terms of that agreement, you have to cooperate so that they can get money from the other involved parties (the drivers). Since the other drivers have insurance agreements as well, rather that your insurance company actually suing the drivers, they approach the other insurance companies, presenting a claim to them (which they will pay on behalf of their client). Of course, those insurance companies can reject the claim (for instance if they think that they are not liable for the damage), but that is risky because that (sort of) leaves your insurance company with no choice other than to sue the other drivers. If that happens, the drivers will in turn sue the insurance companies for breach of contract, and probably file a complaint with the state insurance commission which would not be good for the company.
To sum it up, this is between you and your insurance company, not you and their insurance company. If you "settle" with one or both of the other companies, that means you will have given away any legal leverage that your company would have over the other companies (the other guys give you money, you say "okay, I won't sue the customer", and you probably have breached your agreement with your own insurance, by releasing the other guy from liability). A first step would be to pass the buck to your insurance company – they may approve this plan (I can't imagine why, though), more likely they will not.