Should I accept the insurance company’s first settlement offer?

Do not accept or reject an offer based only on the fact that it is the first one. Compare it with the evidence of liability, completed and future medical care, wage loss, pain and limitations, available coverage, liens, case expenses, and litigation risk. Early offers can arrive before the full injury is known or before all responsible parties and policies have been identified.
Most settlements require a release that permanently ends the claim, even if symptoms later worsen. Ask counsel to explain the projected net recovery and what rights the release gives up. A fair decision depends on the complete record, not pressure to resolve the matter quickly.