Why are you so worried about your children coping with change if you yourself have introduced a new partner into your life and moved house and now have a baby to take care of? Not just any new partner but one who doesn't see the children as part of his remit even though you are all living in the same house.
You seem to have been counting on your ex to facilitate all the changes you wanted and foisted on your children, and you resent him for making a change of his own. He is not the problem here.
You and your P need to sort this out together.
He needs to roll up his sleeves and get more involved in the care and management of the children. I assume you are the one cleaning and cooking and doing laundry and keeping track of everything for everyone now, and this obviously will have to change.
You need to start your children in the local schools and stop complaining about your ex turning the children's world upside down.
What kind of things should I be asking my partner to do? He's pretty good when asked, he doesn't show much initiative though. He says he finds it difficult to know his place in the household.
Meal planning once a week with everyone sitting around the table, P writing the shopping list and the daily meal plan.
His special job can be deep cleaning the bathroom weekly without being asked, using the right cleaning equipment and sprays, etc. Daily once over of the loo and sink.
Getting to grips with school calendar and extra curricular activities and he can be responsible for entering all items into the family calendar (that should be in a prominent place in the kitchen) and he should keep a google calendar too.
Ferrying DCs to some extra curricular activities. Getting to know other parents and the DCs' friends.
Maybe organising carpooling with other parents to/from activities.
Keeping track of homework projects the DCs may have, reminding them of deadlines without nagging.
That is all doable, and all involves hands-on solving of practical problems.
The real crux is diving into the emotional undercurrents, and tough though it is, this has to be done or the house won't be a home.
You and DP should read a few books on blended families. There are many out there - buy a few and read them and discuss them together. You and he need to do this consciously and not just bumble along because the stakes are high for all the children involved.
He needs to understand that bickering is normal, though if it is constant or gets very personal/hurtful remarks are made it is a sign of unhappiness in the children, and the basic unhappiness needs to be tackled.
You and he could read these:
www.empoweringparents.com/article/6-ways-to-stop-sibling-bickering-and-rivalry/#
www.positiveparentingsolutions.com/parenting/when-children-bicker-stopping-fight-before-it-starts