I just remembered how to fix it.
On your Mac, open the Contacts app and export your own contact info to a vcard. Then delete your Contact info (if you're using iCloud, it will automatically be deleted from your other devices, too -- which is what we want in this case). Then double click the vCard to import it. Then, on your devices, set the newly imported card to be your card. Now when you AirDrop, the icon should not be blank any longer.