What should I do if my PCIe Ethernet card screw hole isn't lined up with the case?

It's usually recommended to secure the card if possible. If the screw hole doesn't match up, there might be an installation issue, or it could just need bending to fit right. Either way, not having it secured isn't ideal for the lifespan of your hardware.

Check that the metal support bracket on your Ethernet card isn't bent; it can happen pretty easily. If it is, you can carefully bend it back into shape by hand or using pliers. Also, make sure the card is sitting snugly in the PCIe slot, with the tab on the bottom edge inside the case—definitely not sticking out. Not securing the card could be risky, as the Ethernet cable could pull on it and possibly disconnect it from the motherboard while your PC is running, leading to potential damage.

It would really help if you could share some pictures. It's often easier to diagnose issues visually. You can upload your images to a site like Imgur and then share the link here.