The bad news is you're stuck with your insurance company if you want any coverage for your pregnancy. The beautiful life you're adding to the world is now a pre-existing condition.
The good news is that although your insurance has a high deductible you haven't been paying for the low-deductible/ co-pay plans other people are paying for, which can top $1000/month. More good news is that you have maternity coverage.
Your plan probably has a FAMILY deductible, look at how much that is because you may hit it and the baby deductible and complications would likely be included. There is a max out of pocket number that is typically shown to you by your agent when you get started with the policy. If you have an agent call him/her first, they typically have a lot of information and can probably tell you how to maximize your benefits. Like, if you do hit the FAMILY deductible your husband should go ahead and have anything looked at he's been thinking needs a look. Chiropractors and massage, lactation specialists, podiatry (anybody need orthodics?) all could possibly be covered at that point, we used baby chiropractic which was great for gas and general fussyness for both of my daughters - once they were straight my life was easier.
Second, your doctor probably knows how best to manipulate the situation with each insurance plan, talk with the insurance person there. The insurance company is a good source of an explanation of your benefits but they will not clue you in on how to get them to pay more. The doctor will likely charge all of their fees to the insurance company when the baby is delivered as somebody else said, that sounds very familiar (although it's been a while for me) they will estimate your costs and set you up on a payment plan too.
Something to watch, the doctor usually gets everything all set up so that they're paid in full or close to it by the time of delivery but whoever submits their billing to the insurance first is the one that gets "applied to deductible" and you get a bill from them. In my case the doctor wasn't the first one to submit for payment so the hospital was applied to deductible, the doctor got paid by the insurance (so they got paid twice, once by me and again by insurance) and then I had to get a refund from the doctor to pay the hospital, just what you want to worry about when you're a new mom.
Usually the emergency deductible is less than the regular deductible, so that should apply in the case of complications but if you've hit your deductible I don't think it applies. For instance, say you have a $1000 deductible for emergencies, you decide to pay your pre-natal visits out of pocket and have a home birth (which would likely be less than your $5000 deductible. Then you have complications and have to be airlifted to the hospital and have an emergency C-section. At that point you should would be paying the $1000 emergency deductible. In the same example say all your pre-natal and home birth expenses are $4500, you should then only have to pay $500 toward the emergency stuff - the emergency deductible is usually designed to make your life easier not get more money out of you.
Pay attention to how things are covered after you hit the deductible too, it may only be 80/20 meaning you could be responsible for 20% even after you hit the deductible up to another amount (typically $10,000). Or it could be 100% after the deductible, which I'm sure you prefer.
It's better with a high deductible plan if the baby is born on January 1st because there are lots of doctor visits the first year. My first daughter was born on 12/12 (early, she was supposed to be born on 1/5) so we did have some deductible issues although at the time we only had co-pays for doctor visits. If well care is covered after your deductible, pay attention to that and get as many of those visits done and paid for before you start a new year. Shots in the doctor's office are expensive.
Also don't worry too much, this stuff is all negotiable. If you're willing to work at it you may be able to get the doctor and/or hospital to take less than your deductible from you and receive most of their payment from insurance. Ask your husband to help if he's any good at that type of thing. If you really struggle with the expenses and have to pay it off after the birth (usually the hospital and anesthesiologist) these expenses are more negotiable after you have gotten the services and it's been 6 months - 2 years and they've seen no payment. Although that could have an adverse affect on your credit (some people care about that and some don't).
If the hospital is willing to sell you additional insurance (rider) that saves you money, great! If you have an agent call him/her and have it looked at first. You don't want to buy something that doesn't help you.