Dr will check for you if baby's head engaged at the birth canal and if your cervix has dilate. Thereafter, they will do membrane sweep. I had my first child (membrane sweep) at 40 weeks and second child (membrane sweep) at 39/40 weeks. Both pregnancy I was already 3cm dilate. The reason for membrane sweep is to help induce labor naturally and gynae will insert finger into your cervix to help loosen the amniotic sac. Once this is done, some mummies will experience slight bleeding, pain, waterbag burst. Then you can proceed to hospital when your contractions gets intense or when your gynae tell you to do so. Because sometimes first child will take awhile to dilate.
Mummy i am having the same symptoms as u and we are on same week! Gynae never offer me membrane sweep, let me know how it goes for you! :)