Quoting Mr's&Family*:" They most likely won't stop contractions. My doctor made it clear he wouldn't. I had DD one day shy of ... [snip!] ... So by then she was on formula and had no interest in the boob. They claimed latching issues, I think differently. Good luck!"
You were definitely pretty lucky. My contractions were treated at 35 weeks with all 3 of my mostly full term deliveries. It's pretty standard at most hospitals that they will stop contractions at 35 weeks unless there is other stuff going on. They'll even give steroid shots that late now.
At 37 weeks, my youngest spent 4 days in the NICU with preemie issues.