What a crummy situation. I have a couple of thoughts. The first is that obviously this is unacceptable. Even if you are going to a 1950s model where it's your job to do all the kid stuff, it was acknowledged then that in these very early years, children are hard to take care of and dads should help. You are still in postpartum recovery! Your energy levels are depleted and you're not getting any sleep. You need some backup. He needs to man up and help YOU, and if it makes it easier for him to think of it that way (taking care of you rather than taking care of the babies) then so be it.
Bottom line is, you DO need rest. I know how brutal it is trying to function on no sleep--it makes you literally crazy. Plus the first time you get a bad cold, which you are more susceptible to being so exhausted, Mr. Fiancee is going to have to take over. So you need to rest and he needs to get with it. You MUST find a friend or relative, even just once a week, to watch them so you can catch up on your rest. Doing it for you means you are doing it for them too.
Kristy's idea of writing him a letter is a good one. I have also found letters to be useful--lets me present a case calmly, and it lets the reader absorb what you're saying without leaping to defend himself.
Humans are not programmed to tolerate infants' crying, but they do seem to have 2 different responses: run away from it or run to do something about it. Encourage him to do the second thing, not the first. In fact, whenever you "catch him" doing something good, give him thanks and encouragement. (That doesn't mean he's doing enough, I know, but every stroke helps.)
My words to you (I wouldn't say it this way to him) are: You are doing your part for this family--doing all of the childcare (and housework) AND studying so you can bring in an income. He needs to go beyond the 8-hour effort, not just until you get back on your feet, but for the long haul. Meanwhile you have been toughing it out, and I'm proud of you!