Still not in labour

I'm a first time mum,currently 37 weeks 3 days.I went to go for a checkup last friday and doctor says that the baby is healthy and the head is already facing down plus baby weight is 2.8 kg.While i was behind the curtains i could hear him talking to the nurse that is inside the room that if i did not deliver within 38 weeks,they have to do c section on me.Is there any cases similar to mine?cause i'm scared that maybe the baby is too big thats why its risky for natural birth or something,plus so far i don't feel like i'm giving birth anytime soon cause baby still happy being inside instead of coming out#firstbaby