Be wary of any agency or family that says they will pay for your medical expenses AND learn the laws of your state FIRST. (Select your state and then State Regulation of Adoption Expenses from
this page.) You also need to research the laws in your state regarding whether open adoption contracts are legally binding or not.
Furthermore, I am a believer that asking the adoptive family to pay for medical expenses goes above and beyond what should be expected of a family. That's my personal belief. I've just recently finished paying off my daughter's birth (2003). It took me forever, and many small, small payments, but it was worth not feeling obligated to place with them.
That said, if you can't afford the birth, contact the hospital NOW and see if they can either set you on a payment plan or perhaps even write off your expenses. (Hospitals have to do so many write offs per year.) Make your calls NOW, as you said, as you're trying to decide what is the best option. Don't wait. Get on the phone and see if there is a feasible option for you.