You only discuss one state - Alaska. . If the judgment was entered in Alaska then Alaska law governs.
Statutes of limitations are irrelevant if a judgment was already entered. Renewal of judgments are as per state (Alaska) law. Consult an attorney licensed in Alaska for the laws pertaining to the renewal of judgments there. In NC, judgments can be renewed after 10 years. Perhaps Alaska has a similar rule. What that means is that the judgment can continue to be enforced, either against any real property in Alaska or personal property, including wage garnishment if allowed, and seizure of your bank accounts unless you bank at a bank that is located only in NC.
You will also have to talk to an Alaska attorney about the judgment and the payment you made. If the judgment was paid in full or settled as per an agreement, then the judgment would be marked as satisfied. While you may have a cancelled check, that proves only that a payment was made not that the judgment was paid. To whom was payment made? The creditor? Or its attorney? Does Alaska have a law requiring a creditor to mark a judgment as satisfied?
All judgments earn interest at the legal rate. Costs to collect and post-judgment interest would continue to accrue if the judgment was not paid in full or there was a written agreement to settle.
I would strongly advise to consult an Alaska attorney as to: (1) the period for renewing judgments; (2) whether Alaska has a law requiring a creditor to mark a judgment as satisfied and the steps to invoke that law; and (3) whether you have sufficient proof that the judgment was paid in full/settled; and (4) to get help to follow it.
Most states have their own statute of limitations that governs how long one has to renew a judgment. In North Carolina, that statute of limitations is 10 years for an "action on a judgment". This means that you have ten years from the date of the judgment to entered to file a new action to "renew" a judgment for another ten years. If the judgment was entered in Alaska, then Alaska law governs.
Secondly, can you show how you paid and/or resolved this debt? Who did you pay? Did you pay the full amount or was it a settlement? Is there any documentation other than the check to show the settlement? A canceled check only shows a payment, not necessarily full payment.
I would suggest contacting an Alaska attorney who can respond to the filing, particularly if payment in full is the issue. One, the attorney can determine if they filed the renewal in time and two, the attorney can assert your defense of payment. If you don't take action, the judgment will be renewed. They can then take the new judgment and file it here in North Carolina and proceed to collect. If you wait until they bring the judgment to North Carolina, the Alaska judgment is still out there and could cause an issue with your credit. Either way, you are likely to need an attorney. Better to attack it in Alaska while you can.