This year, I was on these shots at the beginning of my 16th week up to my 36th week. I did read the pros and cons of these shots and the pros do outweigh the cons. I'm not too crazy about chemicals in my body either, but I was at risk for preterm birth because my 1st baby was born at 30 weeks and had to be in the NICU for two months.
I also had a cerclage put in as well and towards the end of September, was placed on bedrest for about a month. My cerclage was taken out at the beginning of my 36th week (a week ago), no more shots and bedrest (yeah!) and am currently 37 weeks 2 days. 37 weeks to term is the best time to have the baby so the goal is to keep it "cooking" inside as long as possible - it's better for the baby.
Do more research online - I'm on the fence w/your situation since your baby's weight seems to be good, but I'm not a doctor.