I had five babies in the US in the 90s and 00s, four with an OB/GYN practice and one with a MW practice, and in two different hospitals. I recall charges similar to HerRoyalNotness's even back then.
I don't recognise Jackieharris' or LemonDrizzle's description of maternity care at all.
I had one experience of stirrups - after delivery - to allow a good angle for the stitching I needed. I had one epidural -- I would have liked to have at least one other but that hospital was overstretched on the day. One baby came too fast to even sign papers, and the last two I just said what the heck and proceeded without any pain relief bar a shot of phenergen.
It has also been my experience that the care you receive has nothing whatsoever to do with the quality of your insurance. I have given birth with private insurance and as a public patient, and there was no difference in care at all. Private rooms were allocated according to availability in the hospitals I was in, with no wards. Non private rooms had two beds. Both private and otherwise had ensuite bathrooms so clean you could eat your meals off the floor. Food was great and plentiful and nurses were lovely.
No gas and air, but do take advantage of having the epidural cannula put in when offered. It doesn't hurt and then it's in place if you decide to have the pain relief. It is also in place if you wind up needing an emergency CS, and you can be awake to see the baby emerge as they use the cannula for partial anesthesia. There are no medals for endurance.
You may have a drip inserted in the back of your hand or wrist as a matter of form, as it may be hospital policy to put women on fluids for labour. The drip can be used also for pitocin and for pain relief/medication for relief of nausea as labour progresses, and also in case you need a transfusion. I recommend your wrist instead of the back of your hand -- and insist they use novocaine to numb the area before inserting the large bore IV needle.
If you are overdue you are more likely to be induced than to be booked for a CS. How long overdue you go depends on the doctor and of course if any problems are detected you will be advised to take the plunge asap. The last month and any overdue weeks involve lots of testing.
A lot of American hospitals concentrate on good nursing care in the maternity area and this makes a huge difference. I have only my sister's horror stories from Dublin to compare with plus some stories from friends in the UK, but my perception is that my US experience of post natal nursing care left theirs in the dust.
If you move at 29 weeks you will have plenty of time to find neighbours or babysitters to take care of your older children while you are in labour. When I was having DD2 I had DD1 and DS at home and got a teenage neighbour to stay with them for the day (a Saturday) but I had two other neighbours who said 'Just call me and I'll drop everything and take the kids in for the day or night' and they meant it..
Insurance:
Make sure you have specifically maternity and newborn insurance, not just regular health insurance with 'family' coverage as they are often separate.
Find out your deductible. 20% is an average percentage iirc.
Ask if PPO (preferred provider organisation iirc) or HMO (health maintenance organisation) or EPO (exclusive provider organisation). PPO gives you more freedom to choose a provider from an insurance company list or from outside the list. Staying in the network/list means you pay less/more is covered by the insurance company.
HMOs have doctors that are like UK GPs - they refer you to your maternity care provider or other specialists.
EPOs are a mixture - you have to stick to the list but you don't have to get a referral.
You need to find out what the out of pocket limit is too, per individual and for the family, and whether that can be set against the deductible and out of pocket limit for the (often separate) maternity coverage if they are separate.
NB: Ask if maternity coverage will cover a pre-existing pregnancy.