For those of you familiar with FMLA vs. short term disability...?

I know the 2 runs concurrently so it will subsidize some of my pay during fmla. My company actually offers a total of 26 weeks short term disability and 12 weeks fmla. I am about 37 weeks and really don't think I can work anymore. If my doctor writes me a note which she said she already would do, can I start the short term disability first and then my fmla when the baby comes so I still have those 12 weeks protected? I heard it worked like that if doctors write a note saying I am unfit for work.