Will my prior car insurer pay injury claims on a car accident I caused while covered?
Termination of coverage (whether due to expiration, or because you changed insurers, or some other reason) does not end the insurer's obligation to pay covered losses or claims occurring while their policy was in effect. The claim should be paid by your then-insurer.
Changing car insurance in search of a better deal or discount is not uncommon. In fact, it's often recommended, since the longer you are with an insurer, the more they tend to "take you for granted" and less they give you a good deal.
There are other reasons you may change insurers, too. So the question is: what happens if you had an accident while covered by Insurer A, but then changed to insurer B? Does A still have to cover that accident? The short answer to this question is: yes, a prior insurer must cover claims arising during the time period you were covered by them, even if they are no longer your insurer when the claim is submitted or comes up for resolution.
When you buy insurance, you purchase coverage for claims arising while that policy is in force or effect. That has two corollaries relevant to this question:
(1) Your second or later insurer has no responsibility for, or obligation, in regards to claims arising prior to the start date of its policy. So you cannot look to your new insurer for coverage.
(2) The prior insurer is responsible for all claims which did arise while you were covered by them (i.e., while your policy with them was in effect), even if you later changed insurers. If the accident occurred while your policy with the first insurer was in force, they have to cover it (subject to the terms and limitations of the policy, of course.) If they will not pay out (or defend you against) a claim arising during their period of coverage, you can sue them for "breach of contract," or violating their contractual obligations, since an insurance policy is really nothing more or less than a contract.
As said above, the short answer is "yes, a prior insurer must cover claims" arising while they covered you.
However, the longer answer is that if your policy with the prior insurer was voided due to some misrepresentation you made on it or while applying for it--for example, claiming that your car was used for personal use only, while you actually use it for business (e.g., making deliveries), or omitting to list some of the drivers in your household (including particularly the one who had the accident)--they can deny coverage. When a contract, including an insurance policy, is voided for a misrepresentation (fraud), the voiding is generally retroactive back to when the policy began. It is essentially "wiped out" back to the start. So if your policy did not expire or was not terminated for some lawful reason (like switching insurers), but rather was voided due to something improper you did in regards to applying for it, then the former insurer would not have to cover the claim.