I'm sorry. I've been awake at night too, but not due to labor. Just insomnia. It does makes the day difficult.
Try not to think of it as "false" labor though. Your body is still working, doing something! Maybe it's trying to get your babe into the perfect position to birth. Maybe it's doing some early work on the cervix to make things ripe! Your body is working, even if it stops in the morning. Try to rest as much as you can, stay hydrated, and I hope you are holding your new little one very soon!!