The hardest thing about change is the unknown. Right now you need to look after yourself and your babies. So what if you don't get things unpacked and put away right away....take your time, make sure you eat, sleep, drink water, get out for a little exercise, and focus on your kids.
There is nothing you can do about the lies he tells you....his head is so far up his ass, in the fog that he simply lies to "give you and answer and shut you up".....don't let him destroy you.
A good friend of mine told me "the best revenge is to do well"....I always made sure I looked great, had my hair/makeup done, and took advantage of the weight I lost (due to the infidelity diet) when I had to see him....don't give him the satisfaction of seeing you broken down. I'm not saying don't cry, that is a normal thing to do, but it's time for the 180....do not let him see it happen...
Reach out to friends/family for help with the kids so you get a break/rest, and make him live up to his financial responsiblities. Have you seen a lawyer yet? Keep the communication between you strictly about the kids.
Hang in there sweetie....you will get through this...come to SI often, post lots...you will get lots of love, support and great advice here from people who do understand because they have either been there, or are there with you.....HUGS