Adjusting liability insurance claims is particularly difficult because there is a third party involved, the plaintiff, who is under no contractual obligation to cooperate with the insurer and may in fact regard the insurer as a deep pocket. The adjuster must obtain legal counsel for the insured (either inside "house" counsel or outside "panel" counsel), monitor litigation that may take years to complete, and appear in person or over the telephone with settlement authority at a mandatory settlement conference when requested by the judge.
The cheapest car insurance, period, will likely carry the minimum coverage required in your state. In most states, this is liability insurance only, which covers property damage and medical bills for others due to accidents you cause. Some states also require uninsured and underinsured motorist coverage, which pay for your injuries or damage if an at-fault driver doesn’t have enough insurance.