While it can vary by the individual policy procedures of your lender, they should be able to backdate those charges if you're able to show proof of insurance for the duration of the loan. You can request a policy declaration or binder from your insurance company which will show the date of policy inception.
Coverage & Claims
As long as the insurance was active at the time you were cited the digital copy should be taken. Oregon is one of the states that accept digital insurance cards per Oregon Law.
While your old insurance card will show your most recent policy term, a letter of experience will provide your new insurance company with additional details from your old policy. This is usually because your new company is providing a hefty discount for the fact that you were maintaining insurance coverage prior to the start of your new policy.
Unfortunately North Carolina does not allow for digital proof of insurance just yet so you will need to print a hard copy to keep in your vehicle. That could always change in the future, but currently a hard copy is required.
The only real solution in this case would be to purchase a basic auto policy as quickly as possible since the impound lot will not release the vehicle until you can prove that it is insured. Asking for the absolute minimum coverage will at least meet the requirement to take your car home.
As of May 2015, 42 states allowed for digital proof of insurance rather than a paper ID card. That is seven more states than in 2013 so there is potential that within the next few years every state could allow for digital proof.