I had exactly the same problem too, only the top part is oily the rest is dry. I did not apply conditioner on the top part but it doesn't help a lot until now I am 31 weeks and the problem has gone, don't worry.
Yes, I have same problem with you, Head OIL very easy to come out, even I wash my hair everyday. But my mum told me no need to wash hair everyday, maybe two days per one.
I am afraid after born baby, how to wait until 12days later, and no wash hair ??? is it bad smell??? I am afraid very dirty.....