Wow, he wants you to just move out with two kids one on the way? Why? Because he said so? Um, no.
Is this because he can't sustain two homes financially?
I wouldn't just up and move in with your mom. Let him move in with a buddy with little or no rent. You and your kids deserve a place that is HOME. Those kids deserve as much consistency as humanly possible right now and that home and their beds is IT.
How often does H have the kids?
Have you considered part time work?
20 hours a week at Starbucks gets you and your family full medical, dental and vision coverage, 1 lb of coffee a week, tips, decent pay for retail work and flexible hours.
I know it's not the direction you were hoping to go but it might be a good news flash to H how much he'd have to be there for the kids if you were working even part time, you know?
I would absolutely not move out. There are other options and he needs to be thinking about the kids.