Sounds like you're going to have quite a trip!!
You've chosen really good dates, it should be fairly quiet, and hopefully the weather will be very decent. We've been around those dates before and always had a perfect time

I don't know how the changing tables are in the other parks, but in our experience, the best place to go when you need to change nappies (if you can afford to be picky, it's not always possible with little ones) is the baby care center in Main Street USA (it's at the end, on the castle side, to the right, next to Plaza Gardens). You have plenty of room and it's not cold (hopefully not a problem in June). They also have some microwaves and high chairs if you need a quiet place to feed the children (otherwise in any restaurant they can heat up your baby food). All toilets have an area where you can change the babies (always bring something to put the baby on, you basically get a tiled surface) so if you can't make it to the baby center it's OK, but unless it was an emergency we always ended up going there when my daughter was younger.
The ESC is great, we stayed there when my daughter was 2. The hotel is not too far from the parks, you got plenty of time, there are long opening hours and long queues should not be a problem, so if I were you I would personally go back for a little break to the hotel during the day. My daughter was used to sleeping anywhere so she could sleep in the pushchair while we stayed in the parks, but she always slept better (and longer) in the room. Then you can go for coffee and something to eat in the ESC lounge, before going back to the park.
You should also try to see Dreams at least once. You can take turns so that someone stays with the twins while others stay to see Dreams (it's at park closing). It's really worth it!!
VIP fastpasses are wonderful. Even if the park should be fairly quiet, some rides (like Big Thunder Mountain, Peter Pan or Ratatouille) can get longer queues.
Try at least one character meal; we did Inventions with our daughter (but it was just the 2 of us paying for the buffet), for a larger group it may be better to go to Cafe Mickey. It's a great way to meet plenty of characters without too much hassle!!
And of course, have a wonderful time!!