it feels like a really strong menstrual cramp
What do the contractions feel like? Are the menstrual cramps the contractions?
But yer your little one will be fine as you're technically full term after week 37 and just wants to come early :) how exciting!
If you don't make it till the end of the month your baby should be just fine being that you are already 38 weeks. I had both of my boys at 36 weeks and they were both healthy. Good luck!