Just enjoy the card when you get it, OP! Why would it be awkward to receive a late one?
We either don’t send cards or send them last minute. Some years we sent Happy New Year’s cards

My ILs send us cards for Chinese New Year, in February, so everyone receives cards all throughout the winter, basically and it’s great. My parents send us cards in early December, and I love that extended Holiday season. Do you also take down your tree on the 26th? The 12 days of Christmas last until Epiphany on Jan 6th, remember, and people around the world celebrate with a cake on that day!
It concerns me that you feel this issue deserves shaming… you must have grown up in a very rigid household. Try to enjoy life. You only have one.