The documents obtained Thursday by ABC11 say the Youngs shared confidential information from a deposition given by John Edwards with federal prosecutors in violation of a court order.
"Defendants and their counsel showed contempt of this court because court had emphatically warned them to comply with its orders," reads the motion.
The motion asks for a hearing on the issue later this month.
Hunter is suing the Youngs over the ownership of a video purportedly showing her and Edwards having sex.
Hunter worked as a videographer for Edwards' presidential campaign. He initially denied an affair and later denied fathering a child with Hunter. He has since admitted to both.
Edwards has been indicted by a federal grand jury on charges he violated campaign finance laws while trying to cover up the affair.
Edwards testified in a February deposition in the sex tape case. Superior Court Judge Carl Fox ordered that his answers be kept sealed.
The case has been put on hold while the criminal case against Edwards moves forward.
In a court filing obtained by ABC11, Edwards said answering questions under oath would violate his Fifth Amendment right not to incriminate himself.