Generally, you cannot sue the insurance company directly, even though the insurance company must pay the verdict.
If you have been injured in an automobile accident, the law requires the at fault driver to have insurance. If the insurance company refuses to make a reasonable settlement offer, the injured party must sue the at fault driver (but not their insurance company). The insurance company will not be a party to the case, even though the insurance company must pay any verdict rendered in the case.
The jury who hears the case will not be allowed to know that insurance is involved. This is true even though the insurance company will hire a lawyer to represent the at fault driver. At trial, the lawyer for the at fault driver will get to stand in front of the jury and tell them they represent the individual. Thus, the jury will hear no evidence about the existence or the extent of insurance involved.
This fact sometimes leads to a much lower jury verdict. If jurors mistakenly think that an individual must come out of pocket to pay the verdict, the jury may be reluctant to render an award that fully compensates you for all of your damages. However, the law does not allow insurance to be mentioned at trial because it could have the opposite effect - it could make jurors award more money than the claim is actually worth because "insurance is paying for it."
This is yet another reason for you to have an experienced lawyer who can explain to the jury that their sole job is to place a fair value on the damages– not speculate about who pays for them.