I've never seen a case like this, so I can only offer a couple of ideas to try.
First, every SD card has a write-protect feature, a small movable tab on the left edge of the card as you look at the label side of the card. Sliding that tab down toward the bottom of that label write-protects the card, and will make is unusable in the camera. Make sure that tab is toward the top of the label and insert in the camera again to see if this helps. I think it unlikely that you'd have the tab moved on 2 cards, so I don't hold out much hope for this being the issue, but it's worth a try.
If the tab on the card isn't the issue, take the battery out of the camera and keep it out for a minute or two to allow the camera to forget everything in its volatile memory. Then put the battery back in and try the card again.
If that doesn't work, do a two-button reset. Your user manual for the 5600 has the specific 2 buttons to press. This resets everything in the camera to its original factory settings. Then, of course, try the card again.
If none of these help, then sending it for repair seems like the only solution to get that specific camera going again. If that's the case, I suggest you look into replacing the camera with a refurbished one from Nikon, or a used one with a warranty from KEH before sending yours away for repair. Understanding the costs of these alternatives will help you decide how best to spend your money.
Good luck.
