D.K.
Wow, it sounds like something's gotta give. This suggestion might not be popular and of course you don't have to take it, but this is what I would personally do.
I'd ditch the PhD for now. Your kids are only young once. They're little and they need you. You can always finish your education later, but you can't make up for their babyhood later. A lot of kids get "sickly" when they're bounced around too much throughout the week. Maybe your babies are trying to tell you something. Maybe they need a little more stability/time and attention from mom.
I'm not criticizing, please believe me. I don't mean anything by my comments. I'm just thinking about how busy you must be and while I sympathize with wanting to further your education and with wanting to make your mark on the world, I empathize with the children that are sick with no mommy there to care for them. You're probably also doing yourself a disservice. I'm sure you're under an enormous amount of stress. Who knows, it might be good for you to put your ambition on hold for a couple of years and let yourself enjoy your babies while they are young.
I know everyone is different, so I'm only speaking from my own feelings here, but I believe that nothing is more important than nurturing the next generation and that they truly need their mother when they're this young. I feel that their needs should come first and mom's wants should come somewhere down the list after that. I'm talking about needs and wants, two different things. Evaluate their needs and weigh them against your desires and see which way the scales tip.
I hope everything works out for you whatever happens. Have a blessed day!