If you make a claim against the other driver's insurance company . . . .
You don’t have a claim “against the other driver’s insurance company”. (If you want to know why, ask and I’ll explain what an indemnity agreement is).
Which means that the other guy’s insurance company is bargaining with you not because they are directly liable to you, but because if their insured is found legally obligated to pay you damages, they must honor their indemnity agreement and pay it on behalf of their insured customer.
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Threatening to sic a lawyer on their insured is not going to help your cause. They have their own attorney’s that will sic right back.
What I suggest is that you tell the adjuster to dispense with the normal b.s. and put the company’s final offer on the table.
If you think it is fair, then accept it. If not, talk to a good trial lawyer. And I don’t mean one that has his grinning face plastered on billboards and on the back covers of telephone books!