I got an accident due to my fault. It seems the personal injury cost of another driver may exceed my coverage.

IF you own the car and have liability insurance your carrier has a duty to defend you in good faith and to pay any  judgment up to the amount of the policy limit of the insurance coverage you bought. They will supply a lawyer, but generally that lawyer will look out for their interests more than yours. If the car owner is your parents they too may have some liability.

If, like most people trying to save some money you were penny wise and pound foolish, and so bought only the minimum coverage, that amount may not be enough to cover the damages sustained in the accident.

In such a case your auto insurer sometimes offers to throw in the policy limits in exchange for a general release for you. The other side first tries to make sure you really have no other assets that could be seized after a judgment and there is no one else who may be liable and/or willing to kick something to resolve the matter, such as your parents, or a business you work for if the accident happened when you were arguably at work, or the bar you had a drink at under the state dram shop law, or the auto manufacturer or repair shop if there was a defect in the car that arguably contributed to the accident.

Assuming there is no other person or firm liable and no other apparent source of recovery and you really have no assets, and you are not expected to have any for years (the victim's lawyer will look closely and may demand some proof of lack of assets from you) the lawyer or party on the other side sometimes takes the deal.

Why? They know you have no assets they could seize. The also know if they sue on and spend the money to take it to trial and get a judgment you probably could file bankruptcy and wipe out any financial liability above the policy limits.

Sometimes the victim's lawyers may not settle and get a huge judgment hoping you'll be very successful and knowing that unless you do file bankruptcy they can seize most of your assets for years. If they force you into bankruptcy sometimes they try to get you to re-accept the debt post bankruptcy..... Do NOT do that.

While the insurance company will defend you to a point, our advice is get a lawyer to look out for your interests, even if you have to borrow the money from your folks. The lawyer's advice costs a relatively small sum and if s/he gets the insurance company and victim to settle with a release to you, you'd be avoiding big exposure and ruining your credit for years.