Haven't read the thread but I had my first two in the UK (busy London teaching hospital) and my third last October at Morristown Memorial in New Jersey with a midwife practice.
My US experience was fantastic - wonderful antenatal care, amazing modern hospital, great post-natal care (was appalling in the UK!). I had dreaded it as had a very negative view of US maternity care. It was my second vbac (first was an emcs).
I had a private, en suite room with a bed for DH. I stayed the two night standard as it was relaxing and lovely. A dedicated nurse would check regularly and expected to change nappies etc (I did all that myself, much to their amusement!). Hospital provided newborn nappies, clothes, blankets etc and a huge range of things like witch hazel pads for my stitches. A post natal massage the day after I delivered and a 'celebration dinner' for myself and DH. It was like a hotel and perfect and cosy for welcoming our third child.
My advice would be research your options - eg can you access a midwife or would you prefer an OB? You will probably need your notes from your last births so get them before you go (though I managed it all via email with my old hospital). Would you like a home birth? (quite a few of my friends and family have had ones in NYC and NJ - home vbac is illegal in NJ or I'd have done it myself!).
Check hospital policies (eg some of my nearest hospitals still have nurseries and take baby away for first hour for tests etc - I hated the thought of that so chose my hospital accordingly).
The U.S. system is as good as you make it - you have options and are your own best advocate. If you meet an OB and don't like him/her then change to another (same is true when looking for a paediatrician for your older dc). Be aware of your co-pays, for antenatal, birth, postnatal and newborn care. All of ours added up to about $800.
Find out if where you're moving has a local message board or facebook groups you could post on for recs on OBs or midwives.
Childcare is always stressful for older DC. Can you have someone fly over to stay until you have the baby? Or else get a temp nanny, eg for a few months until baby is here. Or hire a doula and have her come with you while DH stays with kids.
Good luck! It is exciting and stressful so please post as much as you want. This board was a great support to me when we moved 2.5 yrs ago.