If you cannot prove you have possession of the card used for the purchase (eg: you lost the card), DL should refund the cost of the ticket to the original card and allow you to present a new card to repurchase the ticket. In theory this could happen at the airport, however at an outstation like LED I probably wouldn't want to rely on the non-DL agent processing this correctly or at all.

This isn't about someone else using your ticket, it's about you using someone else's credit card. (Not saying that's what you did, just that's what the fraud detection algorithms are worried about.)