Hi. I have the same situation. I am about 2 mos. pregnant. I called my ob/gyn and the hospital, if you negotiate with them (just ask) they usually cut a percentage off of their fees for cash-paying patients.
I checked most area hospitals. Parkland had a good rate that included prenatal visits, but you couldn't pick your doctor and would see whoever was there that day. Medical City (Forest/75) had a really good rate ($3100 for hospital stay, doesn't include doctors, anesth, etc). Baylor downtown is about $4600 for cash paying people.
I hope this helps, it's a difficult situation! I think most people will work with you. It's becoming more and more common. As far as help paying cash for the discounted rates, I don't know of any programs for that.
C.