That is very odd! I got my OCI from a tourist visa, it took 10 months in India until I got it.
I don't know how long processing time is in TX. But you're probably better off doing there. I'm getting my kids OCI in the states rather than India.
I've done crba in Hyderabad and Delhi and yes they require prenatal records, yes they check them. That's all the doctor visits, ultrasounds and discharge papers.
Are you traveling back to the U.S with the baby after 1 month? You need to get baby's CRBA and passport. Might be longer than a month before you can travel back with the baby.