They are legally obligated to represent you for any accident that occurs during the period they provide coverage for you. So they will continue to represent you and the attorney who handles the case will not even know if you are still insured or not.
Your rights as to insurance coverage are fixed as of the date of your loss, so you may thereafter find new insurance and your claim will not be effected. Whether the insurer continues to fight your case is related to their business practices and/or the facts of your case, not whether you are or are not insured by them after the fact of loss.
No. It's a free market and they are duty bound by to represent your interests whether or no you are presently purchasing insurance from them or some other company.
Your insurance company is required to defend you for an accident which was your fault as long as you had coverage with them which was valid on the date of the accident. No matter what happens after that date, the insurer is still required to indemnify and defend you for that accident.
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