Sex start labor?

So I a little background, I had my first cervical exam yesterday. I was 3 cm and 80% effaced. I came home around 2 pm. And around 9 pm started having contractions. I went to the hospital because I'm a FTM and the were getting harder to breathe through. I was hooked up in triage till 5 am and was told they were just Braxton hicks. They were coming every 5 minutes then 2 minutes then 5 minutes etc. and the strength of them wasn't the same as real labor. She did say the second time she checked me that she could feel the babies head and that she was stationed -1 and the second time she was stationed 0. But because my cervix wasn't dilating anymore, they sent me home. I also lost my mucus plug while at the hospital. I'm not in any spectacular rush to get her out considering I am only 37 weeks. However, I was wondering if you did have sex and walk a lot, how long after the sex did your water break or did you start having regular contractions. Now I know everyone is different, I would just like to hear some stories that's all! Thank you!